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

TL;DR

This paper presents an elementary proof of the Implicit Function Theorem for differentiable maps without assuming continuity of derivatives, using basic calculus tools and avoiding advanced mathematical concepts.

Contribution

It provides a simplified, accessible proof of the theorem under minimal assumptions, extending the classical inverse function theorem.

Findings

Elementary proof of the Implicit Function Theorem

No need for continuity of partial derivatives

Includes a stronger version of the Inverse Function Theorem

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.