Past exam of the options I course of the University of Oxford / 2013 / F / 1 / c / i / Solution

A $C^1$ diffeomorphism $F:U\to V$ is a bijection for which both $F$ and $F^{-1}$ are continuously differentiable. If a higher regularity class is specified, both maps must belong to that class. In particular, a diffeomorphism is a homeomorphism, and the chain rule gives $D(F^{-1})(F(x))=(DF(x){)}^{-1}$.