jorgejasso85xvx
2022-11-19
Answered

Given topological spaces ${X}_{1},{X}_{2},\dots ,{X}_{n},Y$, consider a multivariable function $f:\prod _{i=1}^{n}{X}_{i}\to Y$ such that for any $({x}_{1},{x}_{2},\dots ,{x}_{n})\in \prod _{i=1}^{n}{X}_{i}$, the functions in the family $\{x\mapsto f({x}_{1},\dots ,{x}_{i-1},x,{x}_{i+1},\dots ,{x}_{n}){\}}_{i=1}^{n}$ are all continuous. Must $f$ itself be continuous?

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luluna81mxmbk

Answered 2022-11-20
Author has **17** answers

The answer is "Yes" if you give the product $\prod {X}_{i}$ the sliceю

The answer is "No" if you give the product $\prod {X}_{i}$ the usual product topology. In this case, $f$ is "separately continuous" but not necessarily continuous. The standard example is the function $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined by $f(x,y)=\frac{2xy}{{x}^{2}+{y}^{2}}$ for $(x,y)\ne (0,0)$ and $f(0,0)=(0,0)$. This function is continuous everywhere except $(0,0)$ but is continuous in each variable.

The answer is "No" if you give the product $\prod {X}_{i}$ the usual product topology. In this case, $f$ is "separately continuous" but not necessarily continuous. The standard example is the function $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined by $f(x,y)=\frac{2xy}{{x}^{2}+{y}^{2}}$ for $(x,y)\ne (0,0)$ and $f(0,0)=(0,0)$. This function is continuous everywhere except $(0,0)$ but is continuous in each variable.

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There are 100 two-bedroom apartments in the apartment building Lynbrook West.. The montly profit (in dollars) realized from renting out x apartments is given by the following function.

$P\left(x\right)=-12{x}^{2}+2136x-41000$

How many units should be rented out in order to optimize the monthly rental profit?

What is the maximum monthly profit realizable?

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Use polar coordinates to find the limit. [Hint: Let $x=r\mathrm{cos}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}y=r\mathrm{sin}$ , and note that (x, y) (0, 0) implies r 0.]
$\underset{(x,y)\to (0,0)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}$

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Here

2. Using the concept of limits figure out what the second derivative of

3. Use the theorems of Limits that have been discussed before to show that

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1

2

3

As many as the analyst wants

1

2

3

As many as the analyst wants

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In a regression analysis, the variable that is being predicted is the "dependent variable."

a. Intervening variable

b. Dependent variable

c. None

d. Independent variable

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51.A+2B

52.AB

53. BA