Oscillations, quasi-oscillations and joint continuity
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)].
scientific article; zbMATH DE number 5883046