# Which of the conditions imply that a function is uniformly continuous relative to an uniformity?

Let $$(X, \mathcal{U})$$ be an uniform space. $$f:(X, \mathcal{U})\to (X, \mathcal{U})$$ is called uniformly continuous relative to $$\mathcal{U}$$, if for every entourage $$V\in \mathcal{U}$$, $$(f\times f)^{-1}(V)\in \mathcal{U}$$. It is known that every uniformly continuous function is continuous relative to the uniformly topology and the conversely if $$(X, \mathcal{U})$$ is a compact and $$f$$ is continuous relative to the uniformly topology, then $$f:(X, \mathcal{U})\to (X, \mathcal{U})$$ is uniformly continuous relative to $$\mathcal{U}$$.

Which of conditions for $$f$$ imply that $$f$$ is uniformly continuous?

There are some cases where we have more structure and we get uniform continuity "for free": if $$X$$ is compact (in its topology induced from the uniformity) then any continuous $$f$$ on it to some other uniform space will be uniformly continuous.
If we have a topological vector space in its uniformity induced from some neighbourhood base of $$0$$, the a continuous linear map into another topological vector space is also uniformly continuous.