Analysis: In-Depth Explanation and More!

The proof that if f(x) is continuous at

x = c

, then

\frac{1}{f (x)}

is continuous at

x = c

If f(x) is continuous at

x = c, f (x) \neq 0 then \frac{1}{f (x)}

is continuous at

x = c

.
My proof: for any

ϵ > 0

, we need to show there exists

δ > 0

, such that whenever

| x - c | < δ

, we have

| \frac{1}{f (x)} - \frac{1}{f (c)} | < ϵ

| \frac{1}{f (x)} - \frac{1}{f (c)} | = | \frac{f (x) - f (c)}{f (x) f (c)} |

= | f (x) - f (c) | \frac{1}{| f (x) f (c) |}

= | f (x) - f (c) | \frac{1}{| f (x) |} \frac{1}{| f (c) |}

Since f is continuous at c, thus

| f (x) - f (c) | < ϵ

. I need to find an upper bound of

| f (x) |

.
Because

| f (x) - f (c) | < ϵ

, we then have

| f (c) | = | f (c) - f (x) + f (x) | \leq | f (c) - f (x) | + | f (x) |

< ϵ + | f (x) |

Thus, we find

| f (x) | > | f (c) | - ϵ

. Equivalently,

\frac{1}{| f (x) |} < \frac{1}{| f (c) | - ϵ}

. (Not rigorous here, since

| f (c) | - ϵ

could be negative, how to remedy?)
Finally, (ignore not rigorous part)

| \frac{1}{f (x)} - \frac{1}{f (c)} | < \frac{ϵ}{| f (c) | - ϵ} \frac{1}{| f (c Analysis Answered questionPEEWSRIGWETRYqx 2022-01-14I am trying to do excercise 4.1. in Continuous Time Markov Processes: An Introduction by Thomas Ligett. There I have to provide some counterexample of a function in C ({0, 1}^{N}) . However I dontAnalysis Answered question deiteresfp 2022-01-14 Prove that \frac{d f (c) - c f (d)}{f (d) - f (c)} = \frac{f (ξ)}{f^{'} (ξ)} - ξ for some ξ \in (c, d)1234The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used.Analysis Newton's MethodProduct High School Questions College Questions Math Solver Top Questions Top Questions 2 Company About us Term of Service Privacy Payment Policy Honor Connect with us Contact us Youtube Facebook Get Plainmath App Google Play E-mail us: [email protected] Our Service is useful for: Plainmath is a platform aimed to help users to understand how to solve math problems by providing accumulated knowledge on different topics and accessible examples. 2023 Plainmath. All rights reserved{"serverSide":false,"href":"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/analysis?page=3","location":"\/post-secondary\/calculus-and-analysis\/analysis?page=3","scheme":"https","host":"plainmath.org","port":443,"base":"","pathname":"\/post-secondary\/calculus-and-analysis\/analysis","search":"page=3","sentry_dns":"https:\/\/d4f0b1ed83bf47f0b61da5ebded3a536@sentry.prokit.me\/77","env":"prod"}{"breadcrumbs":[{"name":"College","url":"https:\/\/plainmath.org\/post-secondary"},{"name":"Calculus and Analysis","url":"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis"},{"name":"Analysis","url":"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/analysis"}],"filters":["All","Answered","Unanswered"],"filter":0,"category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"pagination":{"list":[{"id":61056,"h1":"With cash =$2950 and accounts payable at $11,900, what's the","content":"With cash <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c24\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><mjx-c class=\"mjx-c39\"><\/mjx-c><mjx-c class=\"mjx-c35\"><\/mjx-c><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo>=<\/mo><mi mathvariant=\"normal\">$<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>2950<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and accounts payable at $11,900, what's the cash ratio?\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-02-09","isAnswered":true,"url":"\/analysis\/61056-with-cash-equal-2950-and-accounts-payable-at-11-900-whats-the","absoluteUrl":"https:\/\/plainmath.org\/analysis\/61056-with-cash-equal-2950-and-accounts-payable-at-11-900-whats-the","user":{"id":46806,"firstName":"Sage Rodgers","lastName":null,"fullName":"Sage Rodgers ","shortName":"Sage Rodgers ","email":"sage_rodgers@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52250,"h1":"Investigate the convergence or divergence of the sequence\n\na_{1}=\\sqrt{c}, a_{2}=\\sqrt{c+a_{1}}, a_{n+1}=\\sqrt{c+a_{n}},","content":"Investigate the convergence or divergence of the sequence\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom 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class=\"mjx-c221A\"><\/mjx-c><\/mjx-mo><\/mjx-surd><mjx-box style=\"padding-top: 0.185em\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><\/mjx-TeXAtom><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3E\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msub><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>=<\/mo><msqrt><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><\/mrow><\/msqrt><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>=<\/mo><msqrt><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo>+<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msub><\/mrow><\/msqrt><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>=<\/mo><msqrt><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo>+<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/msub><\/mrow><\/msqrt><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>><\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n<br>\nI proved it is increasing. But I could not prove it is bounded above or unbounded. I guess it is divergence sequence. How can I do this?\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52250-investigate-convergence-divergence-sequence-equal-sqrt-equal-sqrt-equal","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52250-investigate-convergence-divergence-sequence-equal-sqrt-equal-sqrt-equal","user":{"id":28936,"firstName":"oliviayychengwh","lastName":null,"fullName":"oliviayychengwh ","shortName":"oliviayychengwh ","email":"oliviayychengwh@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52319,"h1":"How to determine if a series oscillates\n\n\\sum(\\frac{n^{2}+1}{n^{2}+2})^{n}x^{n} where x is","content":"How to determine if a series oscillates\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-lop\"><mjx-c class=\"mjx-c2211 TEX-S2\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"2\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c28 TEX-S3\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.363em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c29 TEX-S3\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 1.237em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo data-mjx-texclass=\"OP\">\u2211<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mfrac><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> where x is real number.\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52319-how-to-determine-if-a-series-oscillates-sum-frac-n-2-plus-1-n-2-plus-where-is","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52319-how-to-determine-if-a-series-oscillates-sum-frac-n-2-plus-1-n-2-plus-where-is","user":{"id":28721,"firstName":"Marenonigt","lastName":null,"fullName":"Marenonigt ","shortName":"Marenonigt ","email":"marenonigt@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52316,"h1":"Convergence of the function series \\sum\\frac{n!}{(nx)^{n}} for x<0","content":"Convergence of the function series <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-lop\"><mjx-c class=\"mjx-c2211 TEX-S2\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c21\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.477em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><mo>!<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> for <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52316-convergence-of-the-function-series-sum-frac-n-nx-n-for-x-less-then-0","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52316-convergence-of-the-function-series-sum-frac-n-nx-n-for-x-less-then-0","user":{"id":29432,"firstName":"percibaa8","lastName":null,"fullName":"percibaa8 ","shortName":"percibaa8 ","email":"percibaa8@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":3,"title":"Female 2","src":"\/build\/images\/user-picture\/user-3.png","src2x":"\/build\/images\/user-picture\/user-3@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52268,"h1":"Let f:(a,b)\\rightarrow R be a differentiable map and let x_{0}\\in(a,b)","content":"Let <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3A\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44F TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D445 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mo>:<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>b<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>R<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> be a differentiable map and let <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2208\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44F TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msub><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>b<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> be a critical point. Prove that if there exists <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D6FF TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3E\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u03b4<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>><\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> such that <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2265\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo data-mjx-pseudoscript=\"true\">\u2032<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>\u2265<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> for all <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2208\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D6FF TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"2\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>\u2212<\/mo><mi>\u03b4<\/mi><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2264\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo data-mjx-pseudoscript=\"true\">\u2032<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>\u2264<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> for all <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2208\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"2\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D6FF TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>+<\/mo><mi>\u03b4<\/mi><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> then <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msub><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/msub><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is a local maximum of f.\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52268-let-f-a-b-rightarrow-r-be-a-differentiable-map-and-let-x-0-in-a-b","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52268-let-f-a-b-rightarrow-r-be-a-differentiable-map-and-let-x-0-in-a-b","user":{"id":28737,"firstName":"interdicoxd","lastName":null,"fullName":"interdicoxd ","shortName":"interdicoxd ","email":"interdicoxd@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52313,"h1":"Problem u_{t}-u_{x x}=e^{t} in \\mathbb{R}\\times(0,+\\infty)","content":"Problem <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D461 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D461 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msub><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>t<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mo>\u2212<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>t<\/mi><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> in <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-ds mjx-b\"><mjx-c class=\"mjx-c211D TEX-A\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-cD7\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c221E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">R<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u00d7<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mo>,<\/mo><mo>+<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52313-problem-u-t-u-x-x-equal-e-t-in-mathbb-r-times-0-plus-infty","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52313-problem-u-t-u-x-x-equal-e-t-in-mathbb-r-times-0-plus-infty","user":{"id":29425,"firstName":"Agohofidov6","lastName":null,"fullName":"Agohofidov6 ","shortName":"Agohofidov6 ","email":"agohofidov6@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":3,"title":"Female 2","src":"\/build\/images\/user-picture\/user-3.png","src2x":"\/build\/images\/user-picture\/user-3@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52286,"h1":"Function f with |f| is Lebesgue integrable but f isn't","content":"Function f with <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is Lebesgue integrable but f isn't locally Lebesgue integrable\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52286-function-f-with-f-is-lebesgue-integrable-but-f-isnt","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52286-function-f-with-f-is-lebesgue-integrable-but-f-isnt","user":{"id":28930,"firstName":"Stacie Worsley","lastName":null,"fullName":"Stacie Worsley ","shortName":"Stacie Worsley ","email":"stacie_worsley@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":3,"title":"Female 2","src":"\/build\/images\/user-picture\/user-3.png","src2x":"\/build\/images\/user-picture\/user-3@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52235,"h1":"Show that \\frac{1}{n}\\sum_{j=1}^{\\infty}(1-p_{j})^{n})\\rightarrow0 as n\\rightarrow\\infty","content":"Show that <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-munderover><mjx-over style=\"padding-bottom: 0.192em; padding-left: 0.368em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c221E\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class=\"mjx-lop\"><mjx-c class=\"mjx-c2211 TEX-S2\"><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><mjx-row><mjx-under style=\"padding-top: 0.167em; padding-left: 0.124em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D457 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-under><\/mjx-row><\/mjx-munder><\/mjx-box><\/mjx-munderover><\/mjx-TeXAtom><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45D TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D457 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.477em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/mrow><\/mfrac><\/mrow><mrow data-mjx-texclass=\"ORD\"><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/mrow><\/munderover><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo>\u2212<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>p<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">)<\/mo><\/mrow><mo stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> as <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c221E\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><mo stretchy=\"false\">\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52235-show-that-frac-1-n-sum-j-equal-1-infty-p-rightarrow0-as-rightarrow-infty","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52235-show-that-frac-1-n-sum-j-equal-1-infty-p-rightarrow0-as-rightarrow-infty","user":{"id":28863,"firstName":"John Stewart","lastName":null,"fullName":"John Stewart ","shortName":"John Stewart ","email":"john_stewart@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":4,"title":"Female 3","src":"\/build\/images\/user-picture\/user-4.png","src2x":"\/build\/images\/user-picture\/user-4@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52295,"h1":"Prove that the equation\n\n2^{x}-1=y^{2} has no integer solutions where x>1.","content":"Prove that the equation\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msup><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> has no integer solutions where <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3E\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>><\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52295-prove-that-the-equation-2-equal-has-no-integer-solutions-where-more-than","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52295-prove-that-the-equation-2-equal-has-no-integer-solutions-where-more-than","user":{"id":28908,"firstName":"Wanda Kane","lastName":null,"fullName":"Wanda Kane ","shortName":"Wanda Kane ","email":"wanda_kane@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":7,"title":"Male 3","src":"\/build\/images\/user-picture\/user-7.png","src2x":"\/build\/images\/user-picture\/user-7@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52310,"h1":"Fonction inverse of f(x)=\\sqrt{x+2}+\\sqrt{x} is \\frac{x^{4}-4x^{2}+4}{4x^{2}}","content":"Fonction inverse of <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msqrt space=\"4\"><mjx-sqrt><mjx-surd><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c221A\"><\/mjx-c><\/mjx-mo><\/mjx-surd><mjx-box style=\"padding-top: 0.181em\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-msqrt space=\"3\"><mjx-sqrt><mjx-surd><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c221A\"><\/mjx-c><\/mjx-mo><\/mjx-surd><mjx-box style=\"padding-top: 0.329em\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>=<\/mo><msqrt><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/msqrt><mo>+<\/mo><msqrt><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/msqrt><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.363em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c34\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c34\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.363em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c34\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c34\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>4<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>4<\/mn><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>4<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>4<\/mn><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-15","isAnswered":true,"url":"\/analysis\/52310-fonction-inverse-of-f-x-equal-sqrt-x-plus-2-plus-sqrt-x-is-frac-x-4-4x-2-plus-4x","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52310-fonction-inverse-of-f-x-equal-sqrt-x-plus-2-plus-sqrt-x-is-frac-x-4-4x-2-plus-4x","user":{"id":28831,"firstName":"lugreget9","lastName":null,"fullName":"lugreget9 ","shortName":"lugreget9 ","email":"lugreget9@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52292,"h1":"Comparison of L^{2}\\ \\text{and}\\ L^{1} norms for functionsIn the context","content":"<p>Comparison of <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D43F TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-c61\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><mjx-c class=\"mjx-c64\"><\/mjx-c><\/mjx-mtext><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D43F TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>L<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mtext>\u00a0<\/mtext><mtext>and<\/mtext><mtext>\u00a0<\/mtext><msup><mrow data-mjx-texclass=\"ORD\"><mi>L<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> norms for functions<br>\nIn the context of Lebesgue Integrals, I have came across <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D43F TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>L<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> as the set of measurable functions <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3A\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c5B\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D44E TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"2\"><mjx-c class=\"mjx-c1D44F TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c5D\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"4\"><mjx-c class=\"mjx-c1D436 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo>:<\/mo><mo stretchy=\"false\">[<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">\u2192<\/mo><mi>C<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> that have Lebesgue integrable squares - that is <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.477em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo stretchy=\"false\">\u2192<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is Lebesgue integrable.<\/p>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52292-comparison-of-l-2-text-and-l-1-norms-for-functionsin-the-context","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52292-comparison-of-l-2-text-and-l-1-norms-for-functionsin-the-context","user":{"id":29375,"firstName":"berljivx8","lastName":null,"fullName":"berljivx8 ","shortName":"berljivx8 ","email":"berljivx8@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":1,"title":"Female 1","src":"\/build\/images\/user-picture\/user-1.png","src2x":"\/build\/images\/user-picture\/user-1@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52343,"h1":"A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist i","content":"A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis.\n<br>\na. What is the pmf of the number of granite specimens selected for analysis?\n<br>\nb. What is the probability that all specimens of one of the two types of rock are selected for analysis?\n<br>\nc. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52343-geologist-collected-specimens-basaltic-specimens-granite-geologist","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52343-geologist-collected-specimens-basaltic-specimens-granite-geologist","user":{"id":29363,"firstName":"maduregimc","lastName":null,"fullName":"maduregimc ","shortName":"maduregimc ","email":"maduregimc@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":1,"title":"Female 1","src":"\/build\/images\/user-picture\/user-1.png","src2x":"\/build\/images\/user-picture\/user-1@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52337,"h1":"A differentiable form on an open subset Usubseteq \\mathbb{R}^{2} is","content":"A differentiable form on an open subset <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D448 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2286\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-ds mjx-b\"><mjx-c class=\"mjx-c211D TEX-A\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>U<\/mi><\/mrow><mo>\u2286<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">R<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is a map\r\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D714 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"4\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D714 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"3\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D714 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u03c9<\/mi><mo>=<\/mo><msub><mi>\u03c9<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><mo>+<\/mo><msub><mi>\u03c9<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\r\nfor some <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msub><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D714 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-msub space=\"2\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D714 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-script style=\"vertical-align: -0.15em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msub><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2208\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D436 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em; margin-left: 0.052em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D448 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msub><mi>\u03c9<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mo>,<\/mo><msub><mi>\u03c9<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mrow><\/mrow><\/msub><mo>\u2208<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>U<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\r\nFor example, if <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2208\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D436 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em; margin-left: 0.052em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D448 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mtext class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D714 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D715\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D715\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D715\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D715\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mo>\u2208<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>U<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>,<\/mo><mtext>\u00a0<\/mtext><mi>\u03c9<\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mi>\u2202<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>\u2202<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mfrac><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mi>\u2202<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>\u2202<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mrow><\/mfrac><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is a differential form.\r\n<br>\nMy question is: what is the difference between <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D715\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u2202<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>?\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52337-a-differentiable-form-on-an-open-subset-usubseteq-mathbb-r-2-is","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52337-a-differentiable-form-on-an-open-subset-usubseteq-mathbb-r-2-is","user":{"id":29417,"firstName":"Joan Thompson","lastName":null,"fullName":"Joan Thompson ","shortName":"Joan Thompson ","email":"joan_thompson@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":1,"title":"Female 1","src":"\/build\/images\/user-picture\/user-1.png","src2x":"\/build\/images\/user-picture\/user-1@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52307,"h1":"Analytic iff Holomorphic on open domainss of \\mathbb{C}","content":"Analytic iff Holomorphic on open domainss of <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-ds mjx-b\"><mjx-c class=\"mjx-c2102 TEX-A\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">C<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52307-analytic-iff-holomorphic-on-open-domainss-of-mathbb-c","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52307-analytic-iff-holomorphic-on-open-domainss-of-mathbb-c","user":{"id":28883,"firstName":"Margie Marx","lastName":null,"fullName":"Margie Marx ","shortName":"Margie Marx ","email":"margie_marx@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":7,"title":"Male 3","src":"\/build\/images\/user-picture\/user-7.png","src2x":"\/build\/images\/user-picture\/user-7@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52280,"h1":"Calculate the improper integral\n\n\\int_{0}^{\\infty}\\frac{1}{\\theta}e^{\\cos\\theta}\\sin(\\sin\\theta)d\\theta","content":"Calculate the improper integral\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-msubsup><mjx-mo class=\"mjx-lop\"><mjx-c class=\"mjx-c222B TEX-S2\"><\/mjx-c><\/mjx-mo><mjx-script style=\"vertical-align: -0.896em; margin-left: -0.388em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\" style=\"margin-left: 0.647em\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c221E\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-spacer style=\"margin-top: 1.506em\"><\/mjx-spacer><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msubsup><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D703 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c73\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D703 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D703 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D703 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><msubsup><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/mrow><\/msubsup><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>cos<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mi>\u03b8<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52280-calculate-the-improper-integral-int-infty-frac-theta-theta-theta-theta","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52280-calculate-the-improper-integral-int-infty-frac-theta-theta-theta-theta","user":{"id":28659,"firstName":"Kelly Nelson","lastName":null,"fullName":"Kelly Nelson ","shortName":"Kelly Nelson ","email":"kelly_nelson@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":7,"title":"Male 3","src":"\/build\/images\/user-picture\/user-7.png","src2x":"\/build\/images\/user-picture\/user-7@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52262,"h1":"How do I proof that this is a metric in R?\n\nI've been struggling to proof that:\n\n|\\frac{x}{","content":"How do I proof that this is a metric in R?\n<br>\nIve\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52262-how-do-proof-that-this-is-metric-in-ive-been-struggling-to-proof-that-frac","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52262-how-do-proof-that-this-is-metric-in-ive-been-struggling-to-proof-that-frac","user":{"id":28950,"firstName":"Miguel Reynolds","lastName":null,"fullName":"Miguel Reynolds ","shortName":"Miguel Reynolds ","email":"miguel_reynolds@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52259,"h1":"How can I find the order of the pole z=\\frac{\\pi}{2}\\ \\text{for}\\ f(z)=\\frac{1}{(2 \\log (z))(1-\\s","content":"How can I find the order of the pole <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D467 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D70B TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-c66\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c72\"><\/mjx-c><\/mjx-mtext><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D467 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c6C\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c67\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D467 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D467 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>z<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mi>\u03c0<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mfrac><\/mrow><mtext>\u00a0<\/mtext><mtext>for<\/mtext><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>z<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>z<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>z<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>? I know the answer should be 2, but I cant\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52259-how-can-i-find-the-order-of-the-pole-z-equal-frac-pi-2-text-for-equal-frac-log","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52259-how-can-i-find-the-order-of-the-pole-z-equal-frac-pi-2-text-for-equal-frac-log","user":{"id":28931,"firstName":"Mabel Breault","lastName":null,"fullName":"Mabel Breault ","shortName":"Mabel Breault ","email":"mabel_breault@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52247,"h1":"The proof that if f(x) is continuous at x=c, then 1\/f(x) is continuous at x=c\n\n","content":"The proof that if f(x) is continuous at <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, then <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is continuous at <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\nIf f(x) is continuous at <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mtext class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2260\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-c74\"><\/mjx-c><mjx-c class=\"mjx-c68\"><\/mjx-c><mjx-c class=\"mjx-c65\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mtext><mjx-mtext class=\"mjx-n\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo>,<\/mo><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2260<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mtext>\u00a0<\/mtext><mtext>then<\/mtext><mtext>\u00a0<\/mtext><mfrac><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is continuous at <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n<br>\nMy proof: for any <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3E\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u03f5<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>><\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, we need to show there exists <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D6FF TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3E\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u03b4<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>><\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, such that whenever <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D6FF TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mi>\u03b4<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, we have <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mi>\u03f5<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\nSince f is continuous at c, thus <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mi>\u03f5<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>. I need to find an upper bound of <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n<br>\nBecause <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mi>\u03f5<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, we then have\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2264\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>\u2264<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mi>\u03f5<\/mi><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n<br>\nThus, we find <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3E\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>><\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>\u2212<\/mo><mi>\u03f5<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>. Equivalently, <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>\u2212<\/mo><mi>\u03f5<\/mi><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>. (Not rigorous here, since <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>\u2212<\/mo><mi>\u03f5<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> could be negative, how to remedy?)\n<br>\nFinally, (ignore not rigorous part)\n<br>\n<span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-stretchy-v style=\"height: 2.42em; vertical-align: -0.96em\" class=\"mjx-c7C\"><mjx-ext><mjx-c><\/mjx-c><\/mjx-ext><mjx-mark><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3C\"><\/mjx-c><\/mjx-mo><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D716 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7C\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo><<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mi>\u03f5<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mrow><mo>\u2212<\/mo><mi>\u03f5<\/mi><\/mrow><\/mfrac><\/mrow><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52247-the-proof-that-if-f-x-is-continuous-at-x-equal-c-then-is-continuous-at-equal","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52247-the-proof-that-if-f-x-is-continuous-at-x-equal-c-then-is-continuous-at-equal","user":{"id":28831,"firstName":"lugreget9","lastName":null,"fullName":"lugreget9 ","shortName":"lugreget9 ","email":"lugreget9@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":2,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52244,"h1":"I am trying to do excercise 4.1. in Continuous Time Markov Processes: An Introduction by Thomas Lige","content":"I am trying to do excercise 4.1. in Continuous Time Markov Processes: An Introduction by Thomas Ligett. There I have to provide some counterexample of a function in <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D436 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-lop\"><mjx-c class=\"mjx-c28 TEX-S2\"><\/mjx-c><\/mjx-mo><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c7D\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.477em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D441 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-lop\"><mjx-c class=\"mjx-c29 TEX-S2\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">{<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo data-mjx-texclass=\"CLOSE\">}<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>N<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>. However I dont\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2022-01-14","isAnswered":true,"url":"\/analysis\/52244-trying-excercise-continuous-time-markov-processes-introduction-thomas","absoluteUrl":"https:\/\/plainmath.org\/analysis\/52244-trying-excercise-continuous-time-markov-processes-introduction-thomas","user":{"id":29397,"firstName":"PEEWSRIGWETRYqx","lastName":null,"fullName":"PEEWSRIGWETRYqx ","shortName":"PEEWSRIGWETRYqx ","email":"peewsrigwetryqx@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":null,"isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":3,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":52241,"h1":"Prove that\n\n\\frac{df(c)-cf(d)}{f(d)-f(c)}=\\frac{f(\\xi)}{f'(\\xi)}-\\xi\n\nfor some \\xi\\in(c,d)","content":"Prove that<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D709 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em; margin-left: 0.053em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D709 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D709 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>\u03be<\/mi><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>\u03be<\/mi><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>\u2212<\/mo><mi>\u03be<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>for some <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D709 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2208\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u03be<\/mi><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>\n","category":{"id":557,"name":"Analysis","title":"Get Expert Help and Answers to Your Analysis Questions","url":"\/post-secondary\/calculus-and-analysis\/analysis","alias":"analysis","seoText":"<div>The Math analysis problems represent one of the most challenging problems for college students because the use of analysis is never easy. The best thing you can do is to see the answers to the questions that you have. You can take these analysis examples and examine each line to see how these came about. If you are unsure about something, it is good to start with the problem and pose it in different words. It will help you to achieve success with the logical proceeding as you provide analysis for the changes, alterations, assumptions, or mathematical-statistical data being used. <\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. 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With cash $= $ 2950$ and accounts payable at $11,900, what's the cash ratio?

Investigate the convergence or divergence of the sequence
$a_{1} = \sqrt{c}, a_{2} = \sqrt{c + a_{1}}, a_{n + 1} = \sqrt{c + a_{n}}, c > 0$ .
I proved it is increasing. But I could not prove it is bounded above or unbounded. I guess it is divergence sequence. How can I do this?

How to determine if a series oscillates
$\sum {(\frac{n^{2} + 1}{n^{2} + 2})}^{n} x^{n}$ where x is real number.

Convergence of the function series $\sum \frac{n!}{{(n x)}^{n}}$ for $x < 0$

Let $f : (a, b) \to R$ be a differentiable map and let $x_{0} \in (a, b)$ be a critical point. Prove that if there exists $δ > 0$ such that $f' (x) \geq 0$ for all $x \in (x_{0} - δ, x_{0})$ and $f' (x) \leq 0$ for all $x \in (x_{0}, x_{0} + δ)$ then $x_{0}$ is a local maximum of f.

Problem $u_{t} - u_{x x} = e^{t}$ in $R \times (0, + \infty)$

Function f with $| f |$ is Lebesgue integrable but f isn't locally Lebesgue integrable

Show that $\frac{1}{n} \sum_{j = 1}^{\infty} {(1 - p_{j})}^{n}) \to 0$ as $n \to \infty$

Prove that the equation
$2^{x} - 1 = y^{2}$ has no integer solutions where $x > 1$ .

Fonction inverse of $f (x) = \sqrt{x + 2} + \sqrt{x}$ is $\frac{x^{4} - 4 x^{2} + 4}{4 x^{2}}$

Comparison of $L^{2} and L^{1}$ norms for functions
In the context of Lebesgue Integrals, I have came across $L^{2}$ as the set of measurable functions $f : [a, b] \to C$ that have Lebesgue integrable squares - that is $x \to {| f (x) |}^{2}$ is Lebesgue integrable.

Analytic iff Holomorphic on open domainss of $C$

Calculate the improper integral
$\int_{0}^{\infty} \frac{1}{θ} e^{\cos θ} \sin (\sin θ) d θ$

How do I proof that this is a metric in R?
Ive

How can I find the order of the pole $z = \frac{π}{2} for f (z) = \frac{1}{(2 \log (z)) (1 - \sin (z))}$ ? I know the answer should be 2, but I cant

I am trying to do excercise 4.1. in Continuous Time Markov Processes: An Introduction by Thomas Ligett. There I have to provide some counterexample of a function in $C ({0, 1}^{N})$ . However I dont

Prove that
$\frac{d f (c) - c f (d)}{f (d) - f (c)} = \frac{f (ξ)}{f^{'} (ξ)} - ξ$
for some $ξ \in (c, d)$