## Continuous function from Continuous functions to R

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:

$$d(f,g)=\sqrt{\int_0^1 (f(x)-g(x))^2 dx}$$.

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 $x_n\rightarrow x$ implies that $f(x_n)\rightarrow f(x)$. So interpreting this for a metric space, a function from one metric space to another is continuous if every sequence such that $f_n\rightarrow f$ implies that $F(f_n)\rightarrow F(f)$.

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

Does this proof work? Thanks in advance!

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