# Continuous Function on Compact Space is Uniformly Continuous

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## Theorem

Let $\R^n$ be the $n$-dimensional Euclidean space.

Let $S \subseteq \R^n$ be a compact subspace of $\R^n$.

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

Then $f$ is uniformly continuous on $S$.

## Proof

This theorem requires a proof.In particular: Use Heine-Cantor TheoremYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness