Continuous Function on Closed Interval is Uniformly Continuous

Theorem

Let $\left[{a .. b}\right]$ be a closed real interval.

Let $f: \left[{a .. b}\right] \to \R$ be a continuous function.

Then $f$ is uniformly continuous on $\left[{a .. b}\right]$.

Proof

We have that $\R$ is a metric space under the (usual) Euclidean metric.

We also have from the Heine-Borel Theorem that $\left[{a .. b}\right]$ is compact.

So the result Heine-Cantor Theorem applies.

$\blacksquare$