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On the k-pseudo symmetric and ordinary differentiation

Łazarow E., Turowska M.

Demonstratio Mathematica

2016

Vol. 49, nr 2

155--160

In 1972, S. Valenti introduced the definition of k-pseudo symmetric derivative and has shown that the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is of Lebesgue measure zero. In 1993, L. Zajíček has shown that for a continuous function f, the set of all points, at which f is symmetrically differentiable but no differentiable, is σ-(1 – ε) symmetrically porous for every ε > 0. The question arises: can we transferred the Zajíček’s result to the case of the k-pseudo symmetric derivative? In this paper, we shall show that for each 0 < ε < 1 the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is σ-(1 – ε)-porous.