Global extrema

Global extrema: Let $K\subseteq\mathbb R^n$ be nonempty and compact, and let $f:K\to\mathbb R$ be continuous . Then there exist points $x_{\min},x_{\max}\in K$ such that

In particular, $f$ has a minimum and a maximum on $K$.

On a closed interval $[a,b]$, this theorem ensures the existence of global maximizers and minimizers needed in many basic arguments in differential and integral calculus.