If $f\colon \mathbb{R} \to \mathbb{R}$ is one-to-one and differentiable at $a$ with $f'(a) \ne 0$, must $f^{-1}$ be differentiable at $f(a)$?
Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable