# Proving a set of functions is bounded in an open set

1. Dec 5, 2013

### mahler1

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

Let $X$ be a complete metric space and consider $C(X)$ the space of continuous functions from $X$ to $\mathbb R$ with the metric $d_{\infty}$. Suppose that for every $x \in X$, the set $\{f(x): f \in C(X)\}$ is bounded in $\mathbb R$. Prove that there exist an open set $U \subset X$ and $C>0$ such that $\forall x \in U$ and $\forall f \in C(X)$, $|f(x)|\leq C$.

The attempt at a solution.

I am totally lost with this problem. I am having trouble understanding what I am trying to prove here. Would the proof of this statement mean that $C(X)$ is bounded restricted to some subset of $X$? Can anyone suggest me where to begin? A lot of information is given in the statement: $X$ is complete, the functions are continuous, etc, but I don't know how to properly use all these facts.

2. Dec 5, 2013

### Dick

I don't get what they mean either. If C(X) is the space of all continuous functions X->R, then any constant function f(x)=M for arbitrary M in R is continous. So f(x) can be anything in R. So how can you assume f(x) is bounded for all f in C(X)? I'm not sure the question (whatever it is) has been stated correctly.

3. Dec 5, 2013

### mahler1

You're right, it doesn't make any sense to assume what the exercise tells to assume because of what you've said. I've taken this exercise from an old exam, if I can figure out if there was any correction on the statement, I'll post it. As you've noticed, as it is now, it is incorrect.