On equi-derivatives
type: Item
instance of: scholarly article
title: On equi-derivatives
zbMATH Open document ID: 0879.26020
published in: Real Analysis Exchange
publication date: 1997-12-10T00:00:00Z
review text: A family of locally Henstock-Kurzweil integrable functions \(f_s:\mathbb R \to \mathbb R\), \(s\in S\) is called a family of equi-derivatives at \(x\in \mathbb R\) if for every \(\eta >0\) there is an \(r >0\) such that for every \(h\), \(0<|h|<r\) and \(s\in S\) we have \(|\frac {1}{h}\int_x^{x+h}f_s(t)dt - f_s(x)|< \eta\) and the family is approximately equicontinuous if for every \(\eta >0\) there is a set \(B\in \mathcal T_d\) such that \(x\in B\) and for all \(t\in B\) and \(s\in S\) the inequality \(|f_s(t) - f_s(x)|<\eta\) holds. (\(\mathcal T_d\) is the so called density topology given by measurable sets for which every \(x\in A\) is a density point of \(A\).) Those two concepts are compared in the paper as well as some properties of functions defined on \(\mathbb R^2\) are studied.
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 1095856
zbMATH Keywords: equi-derivative
zbMATH Keywords: approximate equicontinuity
zbMATH Keywords: Baire property
author: Zbigniew Grande
reviewed by: Štefan Schwabik
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1095856