I found the following problem in Pugh's Analysis book:
If $f:(a,b) \rightarrow (c,d)$ is a homeomorphism of class $C^r$ where $1\leq r < \infty$ and $ \forall x\in (a,b)$
we've $f'(x)≠0$ then $f$ is a $C^r$ diffeomorphism.
My Attempt: Clearly it suffices to prove that $f^{-1}$ is a function of class $C^r .(r\geq1)$
Very naturally I'm trying to do it using induction. For $r=1$ , we can write by Inverse Function Theorem $(f^{-1})'(y)=\frac{1}{f' (f^{-1}(y))}$ and hence $f',f^{-1}$ both are continuous (due to the hypothesis) & we're done.
But for $r\geq2$ , I'm confused a bit regarding the inductive step.
For $r=2$ , it's something like $(f^{-1})''(y)$ = $((f^{-1})'(y))' = (\frac{1}{f' (f^{-1}(y))})'$ but then how to proceed further? And how to proceed inductively for any $r\in \mathbb{N}$.
Maybe it's easy but I really can't get it rightly. If someone can write down the inductive step it'll be helpful for me.
Thanks in advance.
