Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found in the above paper that, $X$ is an Atsuji space if and only if every bounded real-valued continuous function on $X$ is uniformly continuous.
I would like to ask whether the same can be concluded for a uniform space.
That is, can we conclude the following:
For a uniform space $(X,\mathcal U),$ every real-valued continuous function (w.r.t. the topology induced by $\mathcal U$) on $X$ is uniformly continuous if and only if every bounded real-valued continuous function on $X$ is uniformly continuous.
Unfortunately I failed to construct a counterexample and consequently seeking some help.
Hint: Suppose every bounded real-valued continuous function is uniformly continuous. It follows easily that every bounded complex-valued continuous function is uniformly continuous. Now suppose $f$ is a real-valued continuous function and let $g=e^{if}$. Then $g$ is uniformly continuous, and (this is the slightly tricky part that I leave to you) it follows that $f$ is uniformly continuous.
(Show first that if $\epsilon\in(0,1)$ then the definition of uniform continuity is satisfied, then show how the case $\epsilon>0$ follows...)