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Abstract

A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   $lim_{x ↦ x_0} (|f(x)-f(x_0)|)/(|ϕ(x)-ϕ(x_0)|) = 0$ for every $x_0$. G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying   $lim_{t ↦ t_0} (|t-t_0|)/(|ϕ(t)-ϕ(t_0)|) = 0$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.