## Real Analysis Exchange

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

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

#### 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