Level Sets of Differentiable Functions of Two Variables with Non-vanishing Gradient

TL;DR

This paper characterizes the local and global structure of level sets of differentiable functions of two variables with non-zero gradient, showing they are locally homeomorphic to intervals or segments, with a discrete set of special points.

Contribution

It provides a detailed topological description of level sets for such functions, including their local homeomorphism types and global structure, extending understanding of differentiable functions with non-vanishing gradient.

Findings

Level sets are locally homeomorphic to open intervals or segments.

Special points where the structure differs form a discrete set.

Global structure of level sets is also characterized.

Abstract

We show that if the gradient of $f:\RR^2\rightarrow\RR$ exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set $\{x\in\RR^2:f(x)=c\}$ is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point. The second case holds only at the points of a discrete set. We also investigate the global structure of the level sets.