The Implicit Function Theorem for maps that are only differentiable: an elementary proof

Abstract

This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps $F(x,y)$ defined on a finite-dimensional Euclidean space. There are no hypothesis on the continuity of the partial derivatives of $F$. The proof employs determinants theory, the mean-value theorem, the intermediate-value theorem, and Darboux's property (the intermediate-value property for derivatives). The proof avoids compactness arguments, fixed-point theorems, and integration theory. A stronger than the classical version of the Inverse Function Theorem is also shown. An example is given.

12 pages. arXiv admin note: text overlap with arXiv:1312.2445

References in corpus (2)

• The Implicit and the Inverse Function theorems: easy proofs
• The Implicit Function Theorem when the matrix $\frac{\partial F}{\partial y}(x,y)$ is only continuous at the base point