Exercise 1189Topology › Topology of ℝⁿL1Added 3 Nov 2026

Continuous image of a compact set

#compact#continuous image#extrema

Statement

Show that if $K\subseteq\mathbb{R}^n$ is compact and $f:K\to\mathbb{R}$ is continuous, then $f$ attains its maximum and minimum on $K$ .

Hint

$f(K)$ is compact in $\mathbb{R}$ , hence closed and bounded; contains its supremum and infimum.

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Solution

Image compact. $f(K)$ is compact in $\mathbb{R}$ (continuous image of compact is compact).

Extrema. $f(K)$ is closed and bounded in $\mathbb{R}$ : it contains its supremum $M$ and infimum $m$ . So there exist $x_{\max}, x_{\min}\in K$ with $f(x_{\max})=M$ , $f(x_{\min})=m$ .

Conclusion. Continuous functions on compact sets attain their extrema.

Continuous image of a compact set

Statement

Solution

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