Oscillations, quasi-oscillations and joint continuity
type: Item
instance of: scholarly article
title: Oscillations, quasi-oscillations and joint continuity
zbMATH Open document ID: 1216.54005
published in: Annals of Functional Analysis
publication date: 2011-05-02T00:00:00Z
full work available at URL: https://eudml.org/doc/225746
full work available at URL: AFA-tex v1 n2 a14.pdf
review text: The following property of mappings of two variables is introduced. A function \(f:X\times Y\to\mathbb R\) is called quasi-separately continuous at a point \((x_0,y_0)\) if: (1)~\(f_{x_0}\) -- the \(x_0\)-section of \(f\) -- is continuous at \(y_0\), and (2)~for every finite set \(F\subset Y\) and \(\varepsilon>0\) there is an open set \(V\subset X\) such that \(x_0\in \text{cl}(V)\) and \(|f(x,y)-f(x_0,y)|<\varepsilon\) whenever \(x\in V\) and \(y\in F\). \(f\) is quasi-separately continuous provided if it is quasi-separately continuous at each point \((x,y)\in X\times Y\). It is shown that if \(X\) is a separable Baire space and \(Y\) is compact then every quasi-separately continuous function \(f:X\times Y\to\mathbb R\) has the Namioka property, i.e., there exists a dense \(G_{\delta}\)-set \(D\subset X\) such that \(f\) is jointly continuous at each point of \(D\times Y\). To prove this result the author introduces a new version of a topological game and the notion of quasi-oscillation of a function. The game is a modification of the Saint-Raymond game [\textit{J. Saint Raymond}, Proc. Am. Math. Soc. 87, 499--504 (1983; Zbl 0511.54007)].
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 91A44
zbMATH DE Number: 5883046
zbMATH Keywords: Namioka property
zbMATH Keywords: joint continuity
zbMATH Keywords: separate continuity
zbMATH Keywords: quasi-continuity
zbMATH Keywords: quasi-separately continuity
zbMATH Keywords: quasi-oscillation
zbMATH Keywords: topological games
author: Alireza Kamel Mirmostafaee
MaRDI profile type: Publication
OpenAlex ID: W2144266383
DOI: 10.15352/AFA/1399900595
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description: scientific article; zbMATH DE number 5883046