Composition of uniformly continuous functions.
I have seen many posts that seem to suggest that the composition of uniformly continuous functions is necessarily uniformly continuous(on R). However I'm pretty sure this is wrong and I'm hoping we can get a concrete answer. I think I have a counter example. Considering the set $[0,\infty)$, let $f(x)=e^{-x}$ and $g(x)=-x$ then $fg(x)=e^x$ is not uniformly continuous but $f$ and $g$ are.
That's not a counter example, since $g\bigl([0,\infty)\bigr)=(-\infty,0]$ and $f$ is not uniformly continuous on $(-\infty,0]$.