On level sets of a continuous nowhere monotone function

K. Garg

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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 $l i m_{x \mapsto x_{0}} (| f (x) - f (x_{0}) |) / (| ϕ (x) - ϕ (x_{0}) |) = 0$ 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 $l i m_{t \mapsto t_{0}} (| t - t_{0} |) / (| ϕ (t) - ϕ (t_{0}) |) = 0$ . We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.