The function =\sin ^{-1}(\cos x) \) is :

The function f(x) = sin^{-1}(cos x) can be expressed as f(x) = π/2 - |x| for x near 0. At x=0, f(0)=π/2. Left-hand limit: lim_{x→0-} f(x) = π/2, right-hand limit: lim_{x→0+} f(x) = π/2, so f is continuous at 0. However, left derivative = 1, right derivative = -1, so it is not differentiable at 0.