Boundedness of continuous functions on compact sets

Question

Let $E$ and $F$ be two metric spaces. If $K$ is a compact subset of $E$ then a continuous function $f:K\to F$ is always bounded and reachs its maximum.

What happens if we replace $K$ by a closed bounded set $C$ ?

Answer 1

Consider $\mathbb R$ with the metric $d(x,y)=\min\{1,|x-y|\}$. Let $f:\mathbb R \to \mathbb R$ be the function $f(x)=x$. Verify that $\mathbb R$ is closed and bounded and notice that $f$ is not bounded.

Remark: The point is that the "correct" notion of boundedness in metric spaces is that of "total boundedness". For metric spaces a set is compact iff it is complete and totally bounded.