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2016 The Implicit Function Theorem when the Partial Jacobian Matrix is only Continuous at the Base Point
Oswaldo de Oliveira
Real Anal. Exchange 41(2): 377-388 (2016).

Abstract

This article presents an elementary proof of the Implicit Function Theorem for differentiable maps $F(x,y)$, defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the case of a single scalar equation this continuity hypothesis is not required. A stronger than usual version of the Inverse Function Theorem is also shown. The proofs rely on the mean-value and the intermediate-value theorems and Darboux’s property (the intermediate-value property for derivatives). These proofs avoid compactness arguments, fixed-point theorems, and integration theory.

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Oswaldo de Oliveira. "The Implicit Function Theorem when the Partial Jacobian Matrix is only Continuous at the Base Point." Real Anal. Exchange 41 (2) 377 - 388, 2016.

Information

Published: 2016
First available in Project Euclid: 30 March 2017

zbMATH: 1384.26039
MathSciNet: MR3597327

Subjects:
Primary: 26B10 , 26B12
Secondary: 97140

Keywords: Calculus of Vector Functions , Functions of Several Variables , Implicit Function Theorems , Jacobians

Rights: Copyright © 2016 Michigan State University Press

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Vol.41 • No. 2 • 2016
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