On Whitney pairs

A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   ${\displaystyle \underset{x\mapsto x_0}{lim}\frac{|f\left(x\right)-f\left(x_0\right)|}{|\varphi \left(x\right)-\varphi \left(x_0\right)|}=0}$ for every ${\displaystyle x_0}$. 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   ${\displaystyle \underset{t\mapsto t_0}{lim}\frac{|t-t_0|}{|\varphi \left(t\right)-\varphi \left(t_0\right)|}=0}$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.