The function f(x)=cos ^-1(cos x), is

The graph of $f(x)=\cos ^{-1}(\cos x)$ as given in Figure.

It is evident from the curve y = f(x) that f(x) is everywhere continuous but it is not differentiable at $x=n \pi, n \in Z$

Also, $f'(x)=(-1)^n, x \in(n \pi,(n+1) \pi)$, where $n \in Z$

We also observe that f(x) is an even periodic function with period $2 \pi$.