Whitney arcs and 1-critical arcs
Volume 199 / 2008
Abstract
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.
