Extreme value theorem - Imported Knowlpedia Content

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

f(x_{\min}) \leq f(x) \leq f(x_{\max}) \quad \text{for all } x \in K.

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.