(Theorem) Extreme value

Theorem: Extreme value theorem on a metric space

Let $(X,d_X)$ be a metric space and $K\subseteq X$ be a compact subset, and suppose $f:X\to \mathbb{R}$ is a continuous function. Then $f$ attains its maximum and minimum within $K$; that is, there exists $x_0\in K$ such that

$$f(x_0)=\underset{x\in X}{\sup }\{f(x)\}.$$

Proof from Modern Analysis I.

• $f(K)$ is compact since it the image of a compact set under a continuous function
• Since $f(K)$ in $\mathbb{R}$, we can apply the Heine-Borel theorem to conclude that $f(K)$ is closed and bounded, and hence has a least upper bound and greatest lower bound

❓ Question. Why does this hold even if the domain is expanded?

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Examples §

See 2024-10-15