# Metric Spaces of Continuous Function

Let X be a metric space defined as such: $$X = f : [0,1] \to \Re : f \,\text{ is continuous}$$ $$d(f,g) = sup_{x\in[0,1]} | f(x) - g(x)|$$

I need to show:
a) The neighborhood, $N_r (0)$, is uncountably infinite.
b) Let E = { $f \in X : f(0) = 0$}. Prove or disprove E is bounded
c)Prove X is not complete (I can't comment, but yes I need to prove it is not complete)

I am not looking for someone to solve the problem for me, Just to point me in the right direction. I think this is saying that X is a metric space of function (?). I don't believe I've run into this before and don't know how to approach this. Any help would be greatly appreciated.

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$X$ is complete. Surely whoever gave you this problem made a typo. – David Mitra Jan 14 '13 at 1:01

To solve this problem you need to make sure you understand what each and every term means and then translate it to the problem at hand. So, for instance, the neighborhood $N_r(0)$ means (I'm guessing here) the set $\{f\in X \mid d(f,0) < r \}$. So, for the case at hand, $0$ probably means the constantly $0$ function. Further, $d(f,0)$ is, by definition, $\sup_x|f(x)|$. So, $N_r(0)=\{f\in X\mid \sup_x |f(x)|<r\}$.
For every $0<\epsilon <r$ the function $f(x)=\epsilon$ thus satisfies that $f(x)\in N_r(0)$ and there are uncountably many such functions. Similarly, you proceed with the other items (at least those that are correct).