Is a function $f\colon C\to\mathbb{R}$ bounded if $C$ is compact but $f$ is not necessarily continuous?

Question

If I have a function mapping a compact set to the real numbers, is that function bounded? I know that this is true if the function is continuous. But is it true even if the function is not continuous? If so, how do I prove this?

Answer 1

Consider $f: [0, 1] \to \mathbb{R}$ defined by $f(x) = \frac{1}{x}$ if $x \neq 0$ and $f(0) = 2$. Then $f$ is unbounded but its domain $[0,1]$ is compact.