### On Whitney pairs

Marianna Csörnyei (1999)

Fundamenta Mathematicae

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