The Implicit Function Theorem when the Partial Jacobian Matrix is only Continuous at the Base Point

Abstract

This article presents an elementary proof of the Implicit Function Theorem for differentiable maps $F(x,y)$, defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the case of a single scalar equation this continuity hypothesis is not required. A stronger than usual version of the Inverse Function Theorem is also shown. The proofs rely on the mean-value and the intermediate-value theorems and Darboux’s property (the intermediate-value property for derivatives). These proofs avoid compactness arguments, fixed-point theorems, and integration theory.