The function $f(x)=\sin ^{-1}(\sin x)$, is

continuous and differentiable at x = 0

The graph of the function $f(x)=\sin ^{-1}(\sin x)$ is as shown in Figure.

It is evident from the graph of f(x) that f(x) is everywhere continuous but not differentiable at $(2 n+1) \frac{\pi}{2}, n \in Z$.

Also, f(x) is an odd function such that

$f'(x)=(-1)^n, \text { if }(2 n-1) \frac{\pi}{2}<x<(2 n+1) \frac{\pi}{2}$