Continuous function from Continuous functions to R

In summary, the conversation discusses the continuity of a function in a metric space and uses the epsilon-delta definition of continuity to analyze it. The person asking the question provides their answer and reasoning, and the other person confirms that it is correct.
  • #1
PingPong
62
0
Hi,

I think I've gotten this problem, but I was wondering if somebody could check my work. I still question myself. Maybe I shouldn't, but it feels better to get a nod from others.

Homework Statement


Consider the space of functions C[0,1] with distance defined as:

[tex]d(f,g)=\sqrt{\int_0^1 (f(x)-g(x))^2 dx}[/tex].

Suppose we have a function F(f)=f(0). Is this function continuous?


Homework Equations


Epsilon-delta definition of continuity.


The Attempt at a Solution


My answer is no, and my reasoning is based on a bit of real analysis that I remember that I'm trying to apply to metric spaces. In real analysis, a function f is continuous if every sequence [itex]x_n\rightarrow x[/itex] implies that [itex]f(x_n)\rightarrow f(x)[/itex]. So interpreting this for a metric space, a function from one metric space to another is continuous if every sequence such that [itex]f_n\rightarrow f[/itex] implies that [itex]F(f_n)\rightarrow F(f)[/itex].

So, I consider the sequence of functions [itex]f_n(x)=(1-x^2)^n[/itex] whose elements are certainly in C[0,1], and the function [itex]f(x)=0[/itex] which is also in C[0,1]. Now, [itex]d(f,f_n)\rightarrow 0[/itex], so the sequence converges to f. But on the other hand, [itex]F(f_n)=f_n(0)=1\ne F(f)=0[/itex]. Thus the function F is not continuous.

Does this proof work? Thanks in advance!
 
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  • #2
Yes, I think that works.
 
  • #3
Thank you very much!
 

1. What is a continuous function?

A continuous function is a type of mathematical function that has no sudden jumps or breaks in its graph. This means that the function is defined and has a value at every point along its graph.

2. How is continuity defined for a function?

A function is continuous if it satisfies the epsilon-delta definition of continuity, which states that for every small number epsilon, there exists a small enough number delta such that for all x within delta distance of a given point, the function's output will be within epsilon distance of the function's value at that point.

3. What is the importance of continuity in mathematics?

Continuity is important because it allows us to make predictions and draw conclusions about the behavior of a function at points where we do not have explicit values. It also allows us to use methods such as the intermediate value theorem to prove the existence of solutions to equations.

4. Can every function be continuous?

No, not every function can be continuous. A function may have a jump, break, or other type of discontinuity at certain points, making it not continuous. For example, the function f(x) = 1/x is not continuous at x = 0.

5. How can we determine if a function is continuous?

To determine if a function is continuous, we can use the epsilon-delta definition of continuity or check for common types of discontinuities such as jumps, breaks, or asymptotes. We can also use graphical or numerical methods to analyze the behavior of a function and determine its continuity.

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