Question

If f(x)= max{-x-2,-1,x-2}
Tell this function is differentiable and continuous at (1,-1)????

Answer 1

Recall the definition of |x|.
f(x) = |x| - 2 on (-1,1)
otherwise, f(x) = -1
Those sharp vertices make it non-differentiable.
f'(x) = 1 for x<1
f'(x) = 0 for x >= 1
f' is not continuous

Answer 2

What about differential???