On the k-pseudo symmetric and ordinary differentiation

• Autorzy: Łazarow E. , Turowska M.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

k-pseudo symmetric derivative; ordinary derivative; σ-porous set

PL

pochodna k-pseudo symetryczna; zwykła pochodna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2016
• Tom: Vol. 49, nr 2
• Strony: 155--160
• Opis fizyczny: Bibliogr. 3 poz.

Twórcy

autor

Łazarow E.

• Institute of Mathematics, Pomeranian University in Słupsk, ul. Arciszewskiego 22d, 76-200 Słupsk, Poland

autor

Turowska M.

• Institute of Mathematics, Pomeranian University in Słupsk, ul. Arciszewskiego 22d, 76-200 Słupsk, Poland

Bibliografia

• [1] S. Valenti, Sur la dérivation k-pseudo-symétrique des fonctions numériques, Fund. Math. 74 (1972), 147–152.
• [2] L. Zajíček, A note on the symmetric and ordinary derivative, Atti Sem. Mat. Fis. Univ. Modena 41(2) (1993), 263–267.
• [3] L. Zajíček, Porosity and σ-porosity, Real Anal. Exchange 13 (1987–88), 314–350.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.