If $f'$ is periodic , are $f , f'$ uniformly continuous?

$f: ℝ \rightarrow ℝ $ is such that $f'$ exist and $f'$ is periodic. Are $f ,f'$ uniformly continuous ?

Attempt :- I was thinking along a counter example . What if $f'(x)=\tan x$ but then $f(x)=\ln (|sec x|) $ but then it's not becoming continuous at some points .

Is the result actually true? I certainly know some results using existence of $\lim_{x\rightarrow c} f'(x) $ and if the right hand and left hand limit of $f'(x)$ at $x=c$ are not equal , then $f$ is not differentiable at $c$

But these limits may not exist , right ?

Any hints. Thanks .