## Real Analysis Exchange

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

Oswaldo de Oliveira

#### 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 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

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

Dates
First available in Project Euclid: 27 June 2018

https://projecteuclid.org/euclid.rae/1530064971

Digital Object Identifier
doi:10.14321/realanalexch.43.2.0429

Mathematical Reviews number (MathSciNet)
MR3942588

Zentralblatt MATH identifier
06924899

#### Citation

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. https://projecteuclid.org/euclid.rae/1530064971