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

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

2011-09-22

arxiv.org/abs/1109.4959v1

J. Math. Anal. Appl. 270 (2002), no. 2, 369-382

Mathematics

Classical Analysis and ODEs

Scientific paper

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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