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Differentiability points of a distance function. (English) Zbl 1045.26003

Summary: Let \(K\subset[0,1]\) be the usual Cantor set, and let \(A{\overset\text{def} =}\{f\in C(K):0\in\text{Range}(f)\}\). Its distance function \(\varphi:C(K) \to\mathbb{R}\) is defined by \(\varphi(f){\overset\text{def} =} \text{dist}(f,A)\).
In this note we characterize the set of points of Gâteaux differentiability of this function \(\varphi\). We prove that \(\varphi\) is not Gâteaux differentiable at a function \(f\) iff \(Z_f=\{x\in K:f(x)=0\}\) can be covered by disjoint open sets \(U_1,U_2, \dots,U_m\) for which there exist non-zero constants \(c_1,c_2,\dots,c_m\) such that 0 is a porosity point of the set \(\cup^m_{n=1} c_n\text{Range} (f|_{U_n})\).

MSC:

26A16 Lipschitz (Hölder) classes
46G05 Derivatives of functions in infinite-dimensional spaces
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