Continuous attains max/min on compact set - Imported Knowlpedia Content

Continuous attains max/min on compact set: Let $K$ be a compact set and let $f:K\to\mathbb{R}$ be a continuous map. Then there exist points $x_{\min},x_{\max}\in K$ such that $f(x_{\min})\le f(x)\le f(x_{\max})$ for all $x\in K$ .

Equivalent characterizations

Equivalently, the subset $f(K)\subseteq\mathbb{R}$ has both a minimum and a maximum.

Remarks

This can be seen by combining compactness of continuous images with basic order properties of compact subsets of $\mathbb{R}$ .