F continuous at x[sub]0[/sub], prove g is continuous atx[sub]0[/sub]

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In summary, we are trying to prove that if f: E--> R is continuous at x0 and x0 is an element of F contained in E, then g:F--->R defined by g(x)=f(x) for all x elements of F is also continuous at x0. We use the definitions of continuity, |x-x0|<δ and |f(x)-f(x0)|<ε, to prove that |x-x0|<δ and |g(x)-g(x0)|<ε. However, the continuity of g at x0 does not necessarily imply the continuity of f at x0, as shown by a counterexample using step functions.
  • #1
kathrynag
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Homework Statement


Suppose f: E--> R is cont at x0 and x0 is an element of F contained in E. Define g:F--->R by g(x)=f(x) for all x elemts of F. Prove g is continuous at x0. Show by example that the continuity of g at x0 need not imply the continuity of f at x0.

Homework Equations


lx-x0l<delta
lf(x)-f(x0)l<epsilon


The Attempt at a Solution


lx-x0l<delta
lg(x)-g(x0)l<epsilon
lf(x)-f(x0l<epsilon
Ok, then it's continuous because g(x)=f(x)?
 
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  • #2
Your proof isn't very convincing. Write it out in words.
 
  • #3
Morphism's point is that "sketched" the proof but you need to say exactly why those statements prove the theorem.
 
  • #4
Ok, but how would I do the part with showing by example f doesn't need to be continuous. Wouldn't it ahve to be continuous since g(x)=f(x)?
 
  • #5
Do g and f have the same domain...?
 
  • #6
No, so if f is in E, then g could still be continuous?
 
  • #7
F is a subset of E, so we know f is continuous. Hence, for all e>0, there exists d>0 such that
|x-x0<d and x in E implies |f(x)-f(x0|<e

Hence
|x-x0<d and x in F implies |f(x)-f(x0|<e

But f(x)=g(x) and f(x0)=g(x0)

Hence |x-x0<d and x in F implies |g(x)-g(x0|<e

This is a much clearer way of writing the proof

For a counterexample, I would suggest looking at step functions
 
  • #8
Thanks!
 
  • #9
Ok, I'm confused on finding an example. Like what if I choose the function [x]?
 

1. What is the definition of continuity at a specific point?

The definition of continuity at a specific point x[sub]0[/sub] is that for a function f(x), the limit of f(x) as x approaches x[sub]0[/sub] is equal to the value of f(x[sub]0[/sub]). In other words, the function must have a smooth and unbroken graph at x[sub]0[/sub].

2. How is continuity at a point related to the limit of a function?

Continuity at a point x[sub]0[/sub] is directly related to the limit of a function at that point. If a function is continuous at x[sub]0[/sub], it means that the limit of the function at x[sub]0[/sub] exists and is equal to the value of the function at x[sub]0[/sub].

3. What is the epsilon-delta definition of continuity at a point?

The epsilon-delta definition of continuity at a point x[sub]0[/sub] is a precise mathematical way of stating that the limit of a function at x[sub]0[/sub] exists and is equal to the value of the function at x[sub]0[/sub]. It involves using the concepts of epsilon (ε) and delta (δ) to show that for any small positive value of ε, there exists a corresponding small positive value of δ such that if the distance between x and x[sub]0[/sub] is less than δ, then the distance between f(x) and f(x[sub]0[/sub]) is less than ε.

4. What is the relationship between continuity at a point and continuity of a function?

If a function f(x) is continuous at every point in its domain, then it is called a continuous function. This means that the function has a smooth and unbroken graph for all values of x. Continuity at a specific point x[sub]0[/sub] is a necessary condition for a function to be continuous, but it is not sufficient. A function can be continuous at a point but not continuous overall if it has a discontinuity at another point.

5. How can we prove that a function g(x) is continuous at a point x[sub]0[/sub] given that f(x) is continuous at x[sub]0[/sub]?

To prove that g(x) is continuous at x[sub]0[/sub], we can use the definition of continuity and the properties of limits. Since f(x) is continuous at x[sub]0[/sub], we know that the limit of f(x) as x approaches x[sub]0[/sub] is equal to the value of f(x[sub]0[/sub]). From there, we can use algebraic manipulation and the fact that the limit of a product of two functions is equal to the product of their individual limits to show that the limit of g(x) as x approaches x[sub]0[/sub] is also equal to the value of g(x[sub]0[/sub]). Therefore, g(x) is continuous at x[sub]0[/sub].

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