On a property of functions
type: Item
instance of: scholarly article
title: On a property of functions
zbMATH Open document ID: 1107.26004
author: Marcin Grande
published in: Real Analysis Exchange
publication date: 2006-11-13T00:00:00Z
review text: A function \(f: X\to Y\), where \((X,\rho_X)\) and \((Y,\rho_Y)\) are metric spaces, is almost continuous, in the sense of \textit{J. Stallings} [Fundam. Math. 47, 249--263 (1959; Zbl 0114.39102)], iff each set \(U\), open in the product space \(X\times Y\), that contains \(G(f)\), the graph of \(f\), also contains the graph of a continuous function \(g: X\to Y\). Similarly, \(f\) is termed graph continuous iff cl \(G(f)\) contains the graph, \(G(g)\), of a continuous function \(g: X\to Y\). The later concept was introduced by \textit{Z. Grande} [Demonstr. Math. 11, 519--526 (1978; Zbl 0392.26002)] but with a different name. In the present work, the author introduces a new concept of near continuity of \(f: X\to Y\) in the following manner: For each positive real number \(\eta\), let \(A_\eta(f)=\bigcup_{x\in X} (K(x,\eta)\times K(f(x),\eta))\), where \(K(t,\varepsilon)\) denotes the open ball with center \(t\) and radius \(\varepsilon\). Then \(f\) has property (a) iff for each positive \(\eta\), there is a continuous function \(g: X\to Y\) such that \(G(g)\subset A_\eta(f)\). The author observes that the almost continuous functions and the graph continuous functions all have the property (a), but shows that there are functions with closed graphs and satisfying the condition (a) which are neither almost continuous nor graph continuous. Additionally the author establishes several other interesting results, relating to this concept; two of which are: Theorem 2. Let \((X,\rho_X)\) and \((Y,\rho_Y)\) be metric spaces. If each of the functions \(f_n: X\to Y\) has property (a) and if \(f_n\rightrightarrows f\), then \(f\) has property (a). Theorem 7. Let \((X,\rho_X)\) be a separable, dense in itself metric space, and let \((Y,\rho_Y)\) be a metric space. For each function \(f: X\to Y\) there is a sequence of functions \(\{f_n: X\to Y\}^\infty_{n=1}\) such that each \(f_n\) has property (a) and the sequence converges discretely to \(f\).
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 5071506
zbMATH Keywords: almost continuity of Stallings
zbMATH Keywords: graph continuity
zbMATH Keywords: discrete convergence
zbMATH Keywords: uniform convergence
zbMATH Keywords: sum
zbMATH Keywords: product
reviewed by: Robert E. Zink
MaRDI profile type: Publication
full work available at URL: realanalexch.31.2.0469
OpenAlex ID: W1528255308
DOI: 10.14321/REALANALEXCH.31.2.0469
description: scientific article; zbMATH DE number 5071506