A strange criterion for compactness

Question

Is it true that if every continuous real-valued function on a metric space is bounded, then that metric space is compact

Answer 1

Suppose $X$ is not compact, then take a injective sequence $(x_n)_n$ of elements of $X$ such that the values set $X_f := \{ x_n\;|\;n\in\mathbf{N}\}$ is discrete and closed. Then look at the function $f : X_f \to\mathbf{R}$ defined by sending $x_n$ to $n$. This is continuous and not bounded. Now Uryshon's lemma ensures that $f$ can be extended to a continuous function (still noted) $f:X\to\mathbb R$ which won't be bounded.