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.