# Continuous Function on Closed Interval is Uniformly Continuous

## Theorem

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a continuous function.

Then $f$ is uniformly continuous on $\closedint a b$.

## Proof

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

We also have from the Heine-Borel Theorem that $\closedint a b$ is compact.

So the result Heine-Cantor Theorem applies.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $5.8$: Compactness and Uniform Continuity: Proposition $5.8.2$