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Give an example of a noncompact metric space on which every real valued continuous function is uniformly continuous.

My attempt: Consider $\mathbb Q$ with usual metric. As $\mathbb Q$ is not complete it is not compact. And the only continuous function from $\mathbb Q$ to $\mathbb R$ are constant functions so they all are uniformly continuous. Is this correct?

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  • $\begingroup$ From $\Bbb R$ to $\Bbb Q$ there are only constant functions. not the other way round.. $\endgroup$ – Henno Brandsma Aug 8 '20 at 19:38
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Your example is not correct. Any continuous function on the real line restricted to $\mathbb Q$ is continuous.

Instead of this take $X=\mathbb N$ with the usual metric. Every function on this is uniformly continuous.

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