Example of noncompact space in which every real valued continuous function on it is uniformly continuous

Question

I wanted to find

Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous

My attempt:

$X$ is an infinite set $d$ is a discrete metric. Any real-valued continuous function is constant due to totally disconnected ness of X. Which is the trivially uniform continuous function

But As $X=\cup_{x\in X} B(x,1)$ which cannot have finite subcover as each ball contain a single point

Is my attempt is correct?

thanking you

Answer 1

You are right, but for the wrong reason. In such a space, every real valued function is both continuous and uniformly continuous (just take $\delta=1$ in the definition of both, and you're done).