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Zbl 1216.54005
Mirmostafaee, A.K.
Oscillations, quasi-oscillations and joint continuity. (English)
[J] Ann. Funct. Anal. AFA 1, No. 2, 133-138, electronic only (2010). ISSN 2008-8752/e

The following property of mappings of two variables is introduced. A function $f:X\times Y\to\Bbb R$ 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 $f:X\times Y\to\Bbb R$ 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 [{\it J. Saint Raymond}, Proc. Am. Math. Soc. 87, 499--504 (1983; Zbl 0511.54007)].
[Tomasz Natkaniec (Gdańsk)]

MSC 2000:: *54C08 Generalizations of continuity
54C05 Continuous maps
26B05 Continuity and differentiation questions (several real variables)
54C30 Real-valued functions on topological spaces
91A44 Games involving topology or set theory

Keywords: Namioka property; joint continuity; separate continuity; quasi-continuity; quasi-separately continuity; quasi-oscillation; topological games

Citations: Zbl 0511.54007

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