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A note on level sets of differentiable functions \(f(x,y)\) with non-vanishing gradient. (English) Zbl 1401.26025

In this paper, the authors deal with the structure of the level sets of differentiable functions on \(\mathbb{R}^{2}\) with non-vanishing gradient.
It is known that if \(f: \mathbb{R}^{2}\to \mathbb{R}^{2}\) is a differentiable function such that \(|\nabla f(x)|>\eta >0\) for all \(x\) in \(\mathbb{R}^{2}\) then in a neighborhood of its points the level set \(\{x\in \mathbb{R}^{2}: f(x)=c\}\) is homeomorphic to an open interval.
Moreover, under the hypothesis that \(f\) has a non-vanishing gradient, it can happen that in any neighborhood of some of its points the level set is not homeomorphic to an open interval.
The following related result was proved by M. Elekes in [J. Math. Anal. Appl. 270, No. 2, 369–382 (2002; Zbl 1006.26006)]:
If \(f: \mathbb{R}^{2}\to \mathbb{R}\) is a differentiable function with non-vanishing gradient then in a neighborhood of each of its points the level set \(Z=\{x: f(x)=c\}\) is homeomorphic either to an open interval or a union of finitely many open segments passing through a point. Furthermore, the set of branch points is discrete.
In this note, the authors present an alternate proof of this result.

MSC:

26B10 Implicit function theorems, Jacobians, transformations with several variables
26B05 Continuity and differentiation questions

Citations:

Zbl 1006.26006
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