arXiv subjects

Oswaldo R. B. de Oliveira

Publications and source records attributed to Oswaldo R. B. de Oliveira.

2 recordsLinked to original sources

arXiv2017

The Implicit Function Theorem for maps that are only differentiable: an elementary proof

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 hypothesis on the continuity of the partial derivatives of $F$. The proof employs determinants theory, the mean-value theorem, the intermediate-value theorem, and Darboux's property (the intermediate-value property for derivatives). The proof avoids compactness arguments, fixed-point theorems, and integration theory. A stronger than the classical version of the Inverse Function Theorem is also shown. An example is given.

math.CA↗

arXiv2015

The Implicit Function Theorem when the matrix $\frac{\partial F}{\partial y}(x,y)$ is only continuous at the base point

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. 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.

math.CA↗