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Fundamenta Mathematicae

2008 | 199 | 2 | 119-130

Whitney arcs and 1-critical arcs

EN

Abstrakty

EN
A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $lim_{y→x, y∈γ} |f(y)-f(x)|/|y-x| = 0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f'(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.

119-130

wydano
2008

Twórcy

autor
• Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
autor
• Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, U.S.A.
autor
• Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic