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 an 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 some help.*