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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.

We give a geometric characterization of the convex subsets of a Banach space with the property that for any two convex continuous functions on this set, if their sum is Lipschitz, then the functions must be Lipschitz. We apply this result to the theory of Δ-convex functions.

A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $li{m}_{y\to x,y\in \gamma }|f\left(y\right)-f\left(x\right)|/|y-x|=0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This...