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

Proof outline

1. The image $f(K)$ is compact (continuous images of compact sets are compact).
2. In $\mathbb{R}$, compact sets are closed and bounded .
3. By the completeness of $\mathbb{R}$, $f(K)$ has a supremum $M$ and infimum $m$.
4. Since $f(K)$ is closed, $M, m \in f(K)$, so they are attained.

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.