we know that composition two uniformly continous function is uniformly continous.Is the converse true?

Question

Question: we know that composition two uniformly continous function is uniformly continous.Is the converse true? Thought: Since $f(x)= \sin^2(x)$ is uniformly continuous, but $x^2$ is not, the converse is not true. But I am thinking that is there any general setting to that, I mean if $f(g(x))$ is uniformly continuous then can we put some restriction on $f$ or $g$.

Answer 1

If either $f$ or $g$ is constant then $f(g(x))$ is also, so is as uniformly continuous as a function can get. The non-constant one can be as wild as you want.