which points of f(x)=arcsin(sinx) are not differentiable? And what is the range of this function?

Question

How could I do this without graphing? I am just afraid that my test, which does not allow calculators, will have some question asking me to find this information.

Also, when are functions "differentiable"? Are the cusps and corners undifferentiable? Or is it the other way around?

Answer 1

not differentiable if f'(x) is undefined.

f'(x) = 1/√(1-x^2)

so f' is undefined at x=±1

Now review the principal values of arcsin(x)

cusps and corners are not differentiable, since the limit of f' is not the same from both sides.