On strong quasi-continuity of functions of two variables
type: Item
instance of: scholarly article
title: On strong quasi-continuity of functions of two variables
zbMATH Open document ID: 0851.26005
published in: Real Analysis Exchange
publication date: 1996-11-18T00:00:00Z
review text: Let \(E \in \{\mathbb{R}, \mathbb{R}^2\}\). If \(A \subset E\) and \(x \in E\), then \(d_u (A,x)\) denotes the upper outer density of \(A\) at \(x\). (If \(E = R^2\), then we consider the ordinary differentiation basis in \(\mathbb{R}^2.)\) Let \(f : E \to \mathbb{R}\) and \(x \in E\). We will write \(x \in A(f)\) if there is an open set \(U \subset E\) such that \(f |(U \cup \{x\})\) is continuous at \(x\) and \(d_u (U,x) > 0\). We will write \(x \in B(f)\) if for every \(\varepsilon > 0\) we have \(d_u (U,x) > 0\), where \(U = \text{int} \{t \in E : |f(t) - f(x) |< \varepsilon\}\). We will write \(x \in Q_s (f)\) (and say that \(f\) is strongly quasi-continuous at \(x)\) if for every \(\varepsilon > 0\) and every density open set \(U \ni x\) there is an open set \(V\) such that \(V \cap U \neq \emptyset\) and \(|f(t) - f(x) |< \varepsilon\) for \(t \in V \cap U\). We will denote \(A = \{f : A(f) = E\}\), \(B = \{f : B(f) = E\}\), and \(Q_s = \{f : Q_s (f) = E\}\). In the paper under review the author proves several results concerning sets \(A(f)\), \(B(f)\), and \(Q_s (f)\), as well as the families \(A\), \(B\), and \(Q_s\). Main results: 0. For every function \(f\) we have \(A(f) \subset B(f) \subset Q_s (f)\). 1. If \(E \backslash B(f)\) is a null set, then \(f\) is continuous almost everywhere. 2. \(Q_s = B\). 3. The uniform closure of \(A\) is equal to \(Q_s\).
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 878217
zbMATH Keywords: strong quasi-continuity
zbMATH Keywords: strong quasi-equicontinuity
zbMATH Keywords: density topology
author: Zbigniew Grande
reviewed by: Aleksander Maliszewski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 878217