If a function defined on a compact set ${\cal X}$ to ${\mathbb R}$ (with no points undefined), then is this function bounded on ${\cal X}$?

Question

Let ${\cal X}$ be a compact set in Euclidean space, and $f: {\cal X} \to {\mathbb R}$ is a function with $|f(x)| < \infty$ for all $x \in {\cal X}$.

The question is whether we can find a $L \in {\mathbb R}_+$ such that $|f(x)| < L$ holds for all $x \in {\cal X}$?