Q:

Will uniform continuity of $g$ imply the uniform continuity of $f$?

Let $S\subset \mathbb R$ and $g:S\rightarrow \mathbb R$ be uniformly continuous. If $f:S\rightarrow \mathbb R$ be a continuous function such that $f(x)\leq g(x)\forall x\in S$, can we conclude that $f(x)$ is also uniformly continuous on $S$?
Any help will be appreciated

A:

Consider $f(x)=\sin(x^2)$ and $g(x)=1$.

