## Real Analysis Exchange

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

### 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 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

**Permanent link to this document**

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

**Subjects**

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

Secondary: 26A05

**Keywords**

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

#### 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