Example of a non compact metric space

Question

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?

Answer 1

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.