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2018 A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient
Anna K. Savvopoulou, Christopher M. Wedrychowcz
Real Anal. Exchange 43(2): 387-392 (2018). DOI: 10.14321/realanalexch.43.2.0387

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.

Citation

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Anna K. Savvopoulou. Christopher M. Wedrychowcz. "A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient." Real Anal. Exchange 43 (2) 387 - 392, 2018. https://doi.org/10.14321/realanalexch.43.2.0387

Information

Published: 2018
First available in Project Euclid: 27 June 2018

zbMATH: 06924896
MathSciNet: MR3942585
Digital Object Identifier: 10.14321/realanalexch.43.2.0387

Subjects:
Primary: 26A04, 26B10
Secondary: 26B05

Rights: Copyright © 2018 Michigan State University Press

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Vol.43 • No. 2 • 2018
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