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arXiv · 1109.4959

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

Márton Elekes
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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.

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Márton Elekes. 2011-09-22. Level Sets of Differentiable Functions of Two Variables with Non-vanishing Gradient. https://arxiv.org/abs/1109.4959

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