Let $f,g : \mathbb{R} \rightarrow \mathbb{R}$. I want to show uniformly continuity of $f$ under
$f$ is continuous and $f(0) = \frac{\pi}{2}$, $g(x) = \cos(f(x))$ is monotonically decreasing.
I know if $f$ is differentiable and $|f'|$ is bounded then $f$ is uniformly continuous. And from $g(x)$ is monotonically decreasing if $g$ is differentiable, then naively I can guess $g'(x) <0$ (decreasing) and deduce $g'(x) = - f'(x) \sin(f(x))<0$, ...
But the information given here is not enough to show $g$ and $f$ are differentiable.