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If a $C^k$ map $f: X\to Y$ where $X, Y$ are Banach spaces is a diffeomorphism. What can we say about $d_xf$, the differential of $f$ at $x\in X$? It is true that we have $$ \inf_{x\in X}\big|\det [d_xf]\big|>0\,? $$
PS: I know that the converse is true by the Hadamard-Levy theorem: If $f: X\to Y$ is a $C^k$-map and $\exists\epsilon_0>0$ such that $\big|\det [d_xf]\big|>\epsilon_0$, then $f$ is a diffeomorphism.
A counterexample is given by $X=\mathbb{R}$, $Y=\mathbb{R}$, $$ g(x) = x^3+x $$ and $f = g^{-1}$. Since $\sup|g'|=\infty$, it follows that $\inf |f'|=0$.
Generally, one can take any diffeomorphism with unbounded derivative and consider its inverse.
