It is shown that for each k > 1, if f is a Baire one function and f is the product of k bounded Darboux (quasi–continuous) functions, then f is the product of k bounded Darboux (quasi–continuous) Baire one functions as well.

The following two theorems give the flavour of what will be proved. THEOREM. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0, 1] to Y coincide if and only if Y is connected and locally connected. THEOREM. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class fun...

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rati...

We consider two spaces of harmonic functions. First, the space H(U) of functions harmonic on a bounded open subset U of R and continuous to the boundary. Second, the space H0(K) of functions on a compact subset K of R n which can be harmonically extended on some open neighbourhood of K. A bounded open subset U of R is called stable if the space H(U) is equal to the uniform closure of H0(U ). We...

All spaces considered are metric separable and are denoted usually by the letters X, Y , or Z. ω stands for the set of all natural numbers. If a metric separable space is additionally complete, we call it Polish; if it is a continuous image of ω or, equivalently, of a Polish space, it is called Souslin. The main subject of the present paper is the structure of Baire class 1 functions. Recent de...