AA epsilon>0,EEdelta in R,AAx in R(|x−c|<delta=>|f(x)−f(c)|<epsilon) is false?

robbbiehu 2022-10-21 Answered
Let the function f : R R be discontinuous at c. Then the statement: ϵ > 0 , δ R , x R ( | x c | < δ | f ( x ) f ( c ) | < ϵ ) is false. The negation of the statement: ϵ > 0 , δ R , x R ( | x c | < δ a n d | f ( x ) f ( c ) | ϵ ) is false because whenever δ is negative, | x c | < δ is false. Is anything wrong here? Thank you!
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

dkmc4175fl
Answered 2022-10-22 Author has 15 answers
Your original statement is meant to express "continuity at c", but it's not written precisely enough! In fact, the original statement you gave is true, because you can pick δ = 0, regardless of the value of ϵ.
Here's the corrected version of the original statement:
ϵ > 0 δ > 0 x R ( | x c | < δ | f ( x ) f ( c ) | < ϵ )
This statement is actually false (by definition), assuming that f is not continuous at c. The negation is
ϵ > 0 δ > 0 x R ( | x c | < δ   and   | f ( x ) f ( c ) | ϵ )
This statement is actually true.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-04-07
Prove or Disprove: Suppose that f is a real-valued function that is continuous on a nonempty set S in R n and that f ( S ) is compact in R . Then S is a compact set R n .
I just studied the concept of compactness and I am stuck trying to prove/disprove the above statement. Any help is appreciated.
asked 2022-06-26

Given that x<0, find the probability x<1/4 F(x)=a cosπx (-1)/2<x<1/2, 0 otherwise.
I got an answer of 0.3535. Is this correct?

asked 2022-10-20
How do you prove that the function x sin ( 1 x ) is continuous at x=0?
asked 2022-07-08
Let   f   be any function which is defined for all numbers. Show that   g ( x ) = f ( x ) + f ( x )   is even.

(1) e . g .   f ( x ) = x 2 g ( x ) = x 2 + ( x ) 2 = 2 x 2     even

(2) e . g .   f ( x ) = x 3 g ( x ) = x 3 + ( x ) 3 = 0     even

(3) e . g .   f ( x ) = x 3 + x 2         neither even nor odd

(4)   g ( x ) = ( x 3 + x 2 ) + ( x 3 + x 2 ) = 2 x 2 even

But how it can be proven that the claim holds?

And needless to say, can I completely assume "for all numbers" in the problem statements belong to a set of complex numbers(handling imaginary numbers is also required in this problem)?
asked 2022-06-30
Given metric spaces ( X , d ) and ( Y , d ) and continuous mapping S and T from X into Y, prove that the set { x X : S x = T x } is closed in ( X , d ).
I've run out of any ideas where I should start
asked 2022-11-08
let f : R R be a continuous function such that f ( 0 ) 1 and for all real x, f ( x ) 2 3 f ( x ) + 2 0. Prove that f ( x ) 1 for all real x.
I think you would start by factorising to get ( f ( x ) 2 ) ( f ( x ) 1 ) 0 and then f ( x ) 1 but I'm not really sure where to go from there? I think maybe you can use the intermediate value theorem but I'm not exactly sure how. TIA
asked 2022-05-08
Let X, Y and Z be topological spaces. Let f : X Y and g : Y Z such that f is one-to-one. Suppose g f and f 1 are continuous. Can we conclude that g f ( X ) is continuous? If not under what conditions does it hold?

My attempt:
We know that g = ( g f ) f 1 and the domain of f 1 is f ( X ). Since g f and f 1 are continuous, then g is continuous on f ( X ).
I'm not sure if I missed some technicalities on the domains/codomains.