Extreme value theorem

The extreme value theorem states: if $f: K \to \mathbb{R}$ is a continuous function on a compact set $K$, then $f$ attains its maximum and minimum values.

That is, there exist $x_{\max}, x_{\min} \in K$ such that

Classical version

For $f: [a,b] \to \mathbb{R}$ continuous: $f$ attains a maximum and minimum on $[a,b]$.

Counterexamples

• $f(x) = x$ on $(0,1)$ (not closed): no maximum or minimum.
• $f(x) = 1/x$ on $[1, \infty)$ (not bounded): no minimum.