A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient

The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f : ℝ2 → ℝ is a differentiable function with everywhere non-zero gradient then for every point x ∊ ℝ2 in the level set {x : f(x) = c} there is a neighborhood V of x such that {f = c} ∩ V is homeomorphic t...

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Published inReal analysis exchange Vol. 43; no. 2; pp. 387 - 392
Main Authors Savvopoulou, Anna K., Wedrychowcz, Christopher M.
Format Journal Article
LanguageEnglish
Published East Lansing Michigan State University Press 01.01.2018
Subjects
Cattle
Differentiable functions
Diffraction
Implicit functions
INROADS
Mathematical functions
Mathematical theorems
Open intervals
Rectangles
Studies
Technology Acceptance Model
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Summary:The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f : ℝ2 → ℝ is a differentiable function with everywhere non-zero gradient then for every point x ∊ ℝ2 in the level set {x : f(x) = c} there is a neighborhood V of x such that {f = c} ∩ V is homeomorphic to an open interval or the union of finitely many open segments passing through a point. Mathematical Reviews subject classification: Primary: 26B10; Secondary: 26B05 Key words: Implicit Function Theorem, Non-vanishing Gradient, Locally homeomorphic
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0147-1937
1930-1219
DOI:10.14321/realanalexch.43.2.0387