On the almost continuity of the diagonal of functions
type: Item
instance of: scholarly article
title: On the almost continuity of the diagonal of functions
zbMATH Open document ID: 0943.26008
author: Dariusz Doliwa
author: Ryszard Pawlak
published in: Real Analysis Exchange
publication date: 2000-05-21T00:00:00Z
review text: Let \(\mathcal A_0\) be the set of almost continuous mappings of the real line, vanishing at the points of discontinuity. For \(f\in \mathcal A_0\), denote by \(\Delta _{\mathcal A_0}(f)\) the set of \(g\in \mathcal A_0\) such that the diagonal map \(f\Delta g\) defined by \(f\Delta g(x)=(f(x),g(x))\) is almost continuous and vanishes at the points of discontinuity. For \(f\in \mathcal A_0\), the authors give a characterization of \(\Delta _{\mathcal A_0}(f)\), involving the notion of ac-homotopy (cf. \textit{R. J. Pawlak} [Real Anal. Exch. 20, No.~2, 805-814 (1995; Zbl 0835.26004)]). Another result states that if \(f,g\in \mathcal A_0\) and \(f\Delta g\) is almost continuous and vanishes at the points of discontinuity, then \(f+g\), \(f\cdot g\), \(\min (f,g)\), and \(\max (f,g)\) are in \(\mathcal A_0\). The paper contains also some other interesting results.
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 54C08
zbMATH DE Number: 1449436
zbMATH Keywords: diagonal function
zbMATH Keywords: almost continuity
zbMATH Keywords: porosity
reviewed by: Jaroslav Smítal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1449436