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On Baire classification of strongly separately continuous functions. (English) Zbl 1390.54015
The author considers strongly separately continuous functions defined on a subspace of a product of topological spaces $$\{X_t\}_{t\in T}$$ equipped with the Tychonoff topology of pointwise convergence. Let $$X=\prod_{t\in T} X_t$$, $$E\subset X$$, $$a\in E$$, and $$t\in T$$. For $$x\in X$$ define $$x_t^a:T\to\mathbb{R}$$ by $$x^a_t(i)=x(i)$$ for $$i\neq t$$, and $$x^a_t(t)=a(t)$$. A function $$f:E\to\mathbb{R}$$ is strongly separately continuous at a point $$a$$ with respect to the $$t$$-th variable if $$\lim_{x\to a} |f(x)-f(x^a_t)|=0$$. A mapping $$f:E\to\mathbb{R}$$ is strongly separately continuous at a point $$a\in E$$ if $$f$$ is strongly separately continuous at $$a$$ with respect to each variable $$t \in T$$, and $$f$$ is strongly separately continuous on the set $$E$$ if it is strongly separately continuous at every point $$a \in E$$ with respect to each variable $$t\in T$$. Recall that for $$f:\mathbb{R}^n\to\mathbb{R}$$, $$n\in\mathbb{N}$$, the properties of strong separate continuity and continuity are equivalent, see O. Dzagnidze [Real Anal. Exch. 24, No. 2, 695–702 (1999; Zbl 0967.26010)]. The main results of the paper under review are:
(1) If $$X=\prod_{n\in\mathbb{N}}X_n$$ is a product of countably many topological spaces, $$a\in X$$ and $$\sigma(a)$$ is the set of all $$x\in X$$ with $$|\{ n: x_n\neq a_n\}|<\aleph_0$$, then every strongly separately continuous function $$f:\sigma(a)\to\mathbb{R}$$ belongs to the first stable Baire class, i.e. there exists a sequence $$f_i:\sigma(a)\to\mathbb{R}$$ of continuous functions such that for every $$x\in\sigma(a)$$ there is $$i_x\in\mathbb{N}$$ with $$f_i(x)=f(x)$$ for $$i\geq i_x$$.
(2) If $$X$$ is a product of countably many real lines, then there exists a strongly separately continuous function $$f:X\to\mathbb{R}$$ which is not Baire measurable.
(3) If $$X$$ is a countable product of normed spaces, then for any open set $$G\subset\sigma(a)$$ there is a strongly separately continuous function $$f:\sigma(a)\to\mathbb{R}$$ such that $$G$$ is equal to the set of all points $$x\in \sigma(a)$$ at which $$f$$ is discontinuous.

##### MSC:
 54C08 Weak and generalized continuity 54C30 Real-valued functions in general topology 26A21 Classification of real functions; Baire classification of sets and functions
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