If a set $C$ is compact, then all continuous functions on a set $C$ are uniformly continuous.
Does the reverse hold?
Question If all continuous functions on a set $C$ are uniformly continuous, then $C$ is compact?
2026-03-30 05:32:12.1774848732
Reverse of the statement of uniform continuity on a compact
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No. Consider the metric space $(\mathbb{N}, d)$, where $d$ is the Euclidean metric. Then every function is uniformly continuous, but the space is not compact.