Whitney arcs and 1-critical arcs

Marianna Csörnyei¹ ; Jan Kališ² ; Luděk Zajíček³

¹ Department of Mathematics University College London Gower Street, London WC1E 6BT, United Kingdom
² Department of Mathematical Sciences Florida Atlantic University 777 Glades Road Boca Raton, FL 33431, U.S.A.
³ Department of Mathematical Analysis Charles University Sokolovská 83 186 75 Praha 8, Czech Republic

Fundamenta Mathematicae, Tome 199 (2008) no. 2, pp. 119-130.

Résumé

A simple arc $\gamma \subset \mathbb R^n$ is called a Whitney arc if there exists a non-constant real function $f$ on $\gamma$ such that $\lim_{y\to x,\, y\in \gamma}{{|f(y)-f(x) |}/{|y-x|}}=0$ for every $ x\in \gamma$; $\gamma$ is $1$-critical if there exists an $f \in C^1(\mathbb R^n)$ such that $f'(x)=0$ for every $x \in \gamma$ and $f$ is not constant on $\gamma$. We show that the two notions are equivalent if $\gamma$ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc $\gamma$ in $\mathbb R^2$ 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 $n\geq 3$ by the first author in 1999.

Détail

DOI : 10.4064/fm199-2-2

Keywords: simple arc gamma subset mathbb called whitney arc there exists non constant real function gamma lim gamma f y x every gamma gamma critical there exists mathbb every gamma constant gamma notions equivalent gamma quasiarc general simple arcs whitney property weaker example gives arc gamma mathbb each whose subarcs monotone whitney arc which strictly monotone whitney arc answers completely problem petruska which solved geq first author