We establish the following converse of the well-known inverse function theorem. Let g:U→V and f:V→U be inverse homeomorphisms between open subsets of Banach spaces. If g is differentiable of class Cp and f is locally Lipschitz, then the Fréchet derivative of g at each point of U is invertible and f must be differentiable of class Cp.
Publication Source (Journal or Book title)
Journal of Mathematical Analysis and Applications
Lawson, J. (2020). An inverse function theorem converse. Journal of Mathematical Analysis and Applications, 486 (2) https://doi.org/10.1016/j.jmaa.2020.123913