Whitney arcs and 1-critical arcs
Tom 199 / 2008
Fundamenta Mathematicae 199 (2008), 119-130
MSC: Primary 26B05; Secondary 26A30.
DOI: 10.4064/fm199-2-2
Streszczenie
A simple arc 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.
