Real Analysis Exchange

A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient

Anna K. Savvopoulou and Christopher M. Wedrychowcz

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Abstract

The purpose of this note is to give an alternate proof of a result of M. Elekes. We show that if \( f:\mathbb{R}^2\rightarrow \mathbb{R}\) is a differentiable function with everywhere non-zero gradient, then for every point \(x\in \mathbb{R}^2\) in the level set \(\{x\: \:: f(x)=c\}\) there is a neighborhood \(V\) of \(x\) such that \(\{f=c\}\cap V\) is homeomorphic to an open interval or the union of finitely many open segments passing through a point.

Article information

Source
Real Anal. Exchange, Volume 43, Number 2 (2018), 387-392.

Dates
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1530064968

Digital Object Identifier
doi:10.14321/realanalexch.43.2.0387

Mathematical Reviews number (MathSciNet)
MR3942585

Zentralblatt MATH identifier
06924896

Subjects
Primary: 26B10: Implicit function theorems, Jacobians, transformations with several variables 26A04
Secondary: 26B05: Continuity and differentiation questions

Keywords
Implicit Function Theorem Non-vanishing Gradient Locally homeomorphic

Citation

Savvopoulou, Anna K.; Wedrychowcz, Christopher M. A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient. Real Anal. Exchange 43 (2018), no. 2, 387--392. doi:10.14321/realanalexch.43.2.0387. https://projecteuclid.org/euclid.rae/1530064968


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