Real Analysis Exchange

The Implicit Function Theorem when the Partial Jacobian Matrix is only Continuous at the Base Point

Oswaldo de Oliveira

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Real Anal. Exchange, Volume 41, Number 2 (2016), 377-388.

Dates
First available in Project Euclid: 30 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.rae/1490839339

Mathematical Reviews number (MathSciNet)
MR3597327

Zentralblatt MATH identifier
1384.26039

Subjects
Primary: 26B10: Implicit function theorems, Jacobians, transformations with several variables 26B12: Calculus of vector functions
Secondary: 97140

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

Citation

de Oliveira, Oswaldo. The Implicit Function Theorem when the Partial Jacobian Matrix is only Continuous at the Base Point. Real Anal. Exchange 41 (2016), no. 2, 377--388. https://projecteuclid.org/euclid.rae/1490839339


Export citation