Real Analysis Exchange
- Real Anal. Exchange
- Volume 43, Number 2 (2018), 429-444.
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 hypotheses on the continuity of the partial derivatives of \(F\). The proof employs the mean-value theorem, the intermediate-value theorem, Darboux’s property (the intermediate-value property for derivatives), and determinants theory. The proof avoids compactness arguments, fixed-point theorems, and Lebesgue’s measure. A stronger than the classical version of the Inverse Function Theorem is also shown. Two illustrative examples are given.
Real Anal. Exchange, Volume 43, Number 2 (2018), 429-444.
First available in Project Euclid: 27 June 2018
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Mathematical Reviews number (MathSciNet)
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Primary: 26B10: Implicit function theorems, Jacobians, transformations with several variables 26B12: Calculus of vector functions
Implicit function theorems Jacobians Transformations with several variables Calculus of vector functions Implicit Function Theorems Jacobians Transformations with Several Variables Calculus of Vector Functions
de Oliveira, Oswaldo. The Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proof. Real Anal. Exchange 43 (2018), no. 2, 429--444. doi:10.14321/realanalexch.43.2.0429. https://projecteuclid.org/euclid.rae/1530064971