Extreme value theorem

Statement

The extreme value theorem states: if $K$ is a nonempty compact set and $f:K\to\mathbb R$ is a continuous function, 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

If $a\leq b$ , every continuous function $f:[a,b]\to\mathbb R$ attains a maximum and a minimum on $[a,b]$ .

Why compactness matters

The function $f(x)=x$ on $(0,1)$ has neither a maximum nor a minimum.
The function $f(x)=x$ on $[1,\infty)$ has no maximum.

Image formulation

Equivalently, $f(K)\subseteq\mathbb R$ has a minimum and a maximum. The result follows from compactness of continuous images and the order properties of nonempty compact subsets of $\mathbb R$ .