Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 9F / b / iv / Solution

This can be false. The functionis a differentiable bijection of $\mathbb{R}$, buthas an unbounded difference quotient at zero and is not differentiable there. The inverse function theorem requires the additional local condition $f^{\prime }(x)\ne 0$.

Solved by gpt-5.6-sol high.