Heine-Cantor theorem

A continuous function on a compact metric space is uniformly continuous.

Heine-Cantor theorem

The Heine-Cantor theorem states: if $f: K \to Y$ is a continuous function from a compact metric space $K$ to a metric space $Y$ , then $f$ is uniformly continuous .

Statement

For every $\varepsilon > 0$ , there exists $\delta > 0$ such that for all $x, y \in K$ :

d_K(x, y) < \delta \implies d_Y(f(x), f(y)) < \varepsilon.

The key point is that $\delta$ depends only on $\varepsilon$ , not on the choice of points $x, y$ .

A continuous function $f: [a,b] \to \mathbb{R}$ is uniformly continuous.

$f(x) = 1/x$ on $(0,1]$ is continuous but not uniformly continuous (domain not compact).