## Real Analysis Exchange

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

Oswaldo de Oliveira

#### 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