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arXiv · 1708.02065

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

Oswaldo R. B. de Oliveira
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Abstract

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.

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BibTeXRIS

Oswaldo R. B. de Oliveira. 2017-08-04. The Implicit Function Theorem for maps that are only differentiable: an elementary proof. https://doi.org/10.14321/realanalexch.43.2.0429

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