Whitney arcs and 1-critical arcs

• Autorzy: Marianna Csörnyei , Jan Kališ , Luděk Zajíček
• Języki publikacji: 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.

Kategorie tematyczne

• 26B05: Continuity and differentiation questions
• 26A30: Singular functions, Cantor functions, functions with other special properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2008
• Tom: 199
• Numer: 2
• Strony: 119-130

Daty

wydano

2008

Twórcy

autor

Marianna Csörnyei

• Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

autor

Jan Kališ

• Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, U.S.A.

autor

Luděk Zajíček

• Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Identyfikatory

DOI

10.4064/fm199-2-2