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 $x, y \in K$:

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

Proof idea

1. Cover $K$ with open balls where $f$ varies by less than $\varepsilon$.
2. By compactness, extract a finite subcover.
3. Use the Lebesgue number of the cover as $\delta$.

Classical version

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

Counterexample

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