Real Analysis Exchange

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

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

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

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