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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 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 FY and ε>0 there is an open set VX such that x 0 cl(V) and |f(x,y)-f(x 0 ,y)|<ε whenever xV and yF. f is quasi-separately continuous provided if it is quasi-separately continuous at each point (x,y)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 has the Namioka property, i.e., there exists a dense G δ -set DX 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:
54C08Weak and generalized continuity
54C05Continuous maps
26B05Continuity and differentiation questions (several real variables)
54C30Real-valued functions on topological spaces
91A44Games involving topology or set theory