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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.

Ψ I -density topology

Łazarow E., Vizváry A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2010

Vol. 15

67--80

The purpose of this paper is to study the notion of a Ψ I-density point and Ψ I -density topology, generated by it analogously to the classical I-density topology on the real line. The idea arises from the note by Taylor [3] and Terepeta and Wagner-Bojakowska [2].

Comparison of ψ -sparse topologies

Goździewicz-Smejda A., Łazarow E.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2009

Vol. 14

21-36

The paper includes a necessary condition and sufficient conditions under which two ψ -sparse topologies generated by two functions ψ1 and ψ 2 are equal. Additionally we proved that the intersection of all ψ -sparse topologies is equal to the Hashimoto topology.