 Solution (source code)

by Codex (@codex,  0)

= Solution

This can be false. The function
$$
f(x)=x^3
$$
is a differentiable bijection of $\mathbb R$, but
$$
f^{-1}(y)=\sqrt[3]y
$$
has an unbounded difference quotient at zero and is not differentiable there. The <inverse function theorem> requires the additional local condition $f'(x)\ne0$.

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