Real Analysis Exchange

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

Oswaldo de Oliveira

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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


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