Preface. Here is the problem I am working on:
Suppose $I$,$J$ are intervals and a monotone onto $f:I\to J$ has an inverse $g:J\to I$. Suppose $x\in I$ and $y:=f(x)\in J$, and that $g$ is differentiable at $y$. Prove:
- If $g'(y)\ne 0$, then $f$ is differentiable at $x$.
- If $g'(y) = 0$, then $f$ is not differentiable at $x$.
I've been working on this assignment for so long now that I'm at my breaking point. I feel like all I need is a little push in the right direction, that is what kinds of things should I be thinking about to get started with this? I don't want a full solution, just some suggestions and pointers.