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Continuous function from Continuous functions to R

  1. Apr 8, 2008 #1
    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.

    1. The problem statement, all variables and given/known data
    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?


    2. Relevant equations
    Epsilon-delta definition of continuity.


    3. 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!
     
  2. jcsd
  3. Apr 8, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, I think that works.
     
  4. Apr 9, 2008 #3
    Thank you very much!
     
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