## Real Analysis Exchange

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

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

Anna K. Savvopoulou and Christopher M. Wedrychowcz

#### 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