Real Analysis Exchange

The Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proof

Oswaldo de Oliveira

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 hypotheses on the continuity of the partial derivatives of \(F\). The proof employs the mean-value theorem, the intermediate-value theorem, Darboux’s property (the intermediate-value property for derivatives), and determinants theory. The proof avoids compactness arguments, fixed-point theorems, and Lebesgue’s measure. A stronger than the classical version of the Inverse Function Theorem is also shown. Two illustrative examples are given.

Article information

Real Anal. Exchange, Volume 43, Number 2 (2018), 429-444.

First available in Project Euclid: 27 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B10: Implicit function theorems, Jacobians, transformations with several variables 26B12: Calculus of vector functions
Secondary: 26A05

Implicit function theorems Jacobians Transformations with several variables Calculus of vector functions Implicit Function Theorems Jacobians Transformations with Several Variables Calculus of Vector Functions


de Oliveira, Oswaldo. The Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proof. Real Anal. Exchange 43 (2018), no. 2, 429--444. doi:10.14321/realanalexch.43.2.0429.

Export citation