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 in...

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Bibliographic Details
Published inReal analysis exchange Vol. 43; no. 2; pp. 429 - 444
Main Author Oliveira, Oswaldo de
Format Journal Article
LanguageEnglish
Published East Lansing Michigan State University Press 01.01.2018
Subjects
Boundary value problems
Derivatives
Differentiable functions
Differential calculus
Euclidean geometry
Euclidean space
Fixed points (mathematics)
Implicit functions
INROADS
Inverse functions
Jacobians
Mathematical functions
Mathematical theorems
Matrices
Parallelepipeds
Sine function
Theorem proving
Theorems
Topological theorems
Uniqueness
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Summary: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. Mathematical Reviews subject classification: Primary: 26B10, 26B12 Key words: Implicit function theorems, Jacobians, Transformations with several variables, Calculus of vector functions
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0147-1937
1930-1219
DOI:10.14321/realanalexch.43.2.0429