Theorem: If f is continuous and strictly monotonic on [a,b], then f^(-1) is continuous on [f(a), f(b)].

Derivative of inverse: If f is differentiable at x = a with f'(a) != 0, then f^(-1) is differentiable at y = f(a) with (f^(-1))'(y) = 1/f'(a).

When f'(a) = 0: f^(-1) has a vertical tangent at y = f(a) — not differentiable.

Example: f(x) = $x^3$ at x = 0: f'(0) = 0. f^(-1)(y) = y^$\frac{1}{3}$ is not differentiable at y = 0 (vertical tangent).