The Implicit Function Theorem when the matrix $\frac{\partial F}{\partial y}(x,y)$ 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. 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.

9 pages - new version with 10 pages, two added references on section 1, corrected grammar

References in corpus (1)

• The Implicit and the Inverse Function theorems: easy proofs

Cited by in corpus (2)

• The Implicit Function Theorem for maps that are only differentiable: an elementary proof
• New inverse and implicit function theorems for differentiable maps with isolated critical points