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2018 The Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proof
Oswaldo de Oliveira
Real Anal. Exchange 43(2): 429-444 (2018). DOI: 10.14321/realanalexch.43.2.0429

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.

Citation

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Oswaldo de Oliveira. "The Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proof." Real Anal. Exchange 43 (2) 429 - 444, 2018. https://doi.org/10.14321/realanalexch.43.2.0429

Information

Published: 2018
First available in Project Euclid: 27 June 2018

zbMATH: 06924899
MathSciNet: MR3942588
Digital Object Identifier: 10.14321/realanalexch.43.2.0429

Subjects:
Primary: 26B10, 26B12
Secondary: 26A05

Rights: Copyright © 2018 Michigan State University Press

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Vol.43 • No. 2 • 2018
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