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Oscillations, quasi-oscillations and joint continuity. (English) Zbl 1216.54005
The following property of mappings of two variables is introduced. A function $f:X×Y\to ℝ$ is called quasi-separately continuous at a point $\left({x}_{0},{y}_{0}\right)$ 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 $\epsilon >0$ there is an open set $V\subset X$ such that ${x}_{0}\in \text{cl}\left(V\right)$ and $|f\left(x,y\right)-f\left({x}_{0},y\right)|<\epsilon$ whenever $x\in V$ and $y\in F$. $f$ is quasi-separately continuous provided if it is quasi-separately continuous at each point $\left(x,y\right)\in X×Y$. It is shown that if $X$ is a separable Baire space and $Y$ is compact then every quasi-separately continuous function $f:X×Y\to ℝ$ 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 $D×Y$. 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 [J. Saint Raymond, Proc. Am. Math. Soc. 87, 499–504 (1983; Zbl 0511.54007)].
MSC:
 54C08 Weak and generalized 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