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arXiv · 1312.2445

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

Oswaldo R. B. de Oliveira
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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. 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 R. B. de Oliveira. 2013-12-09. The Implicit Function Theorem when the matrix $\frac{\partial F}{\partial y}(x,y)$ is only continuous at the base point. https://doi.org/10.14321/realanalexch.41.2.0377

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