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On the Darboux property of the sum of cliquish functions. (English) Zbl 0762.26001
The starting point of the considerations of this paper is the well-known theorem proved by H. W. Pu and H. H. Pu in 1987 [Čas. Pěstovaní Mat. 112, 320-326 (1987; Zbl 0646.26004)] (For each finite family $$A$$ of Baire 1 functions mapping $$\mathbb{R}$$ into $$\mathbb{R}$$, there exists a Baire 1 function $$f$$ such that $$f+g$$ is a Darboux function for $$g\in A$$).
In this paper the author proves: For each finite family $$A$$ of Baire 1 functions mapping $$\mathbb{R}$$ into $$\overline{\mathbb{R}}$$ $$(=\mathbb{R}\cup\{- \infty,+\infty\})$$ such that $$\{x: h(x)=\pm\infty\}$$ is nowhere dense (for $$h\in A$$), there exists a Baire 1 function $$f: \mathbb{R}\to\mathbb{R}$$ such that $$f+g$$ is a Darboux function for $$g\in A$$. For each finite family $$A$$ of cliquish functions mapping $$\mathbb{R}$$ into $$\mathbb{R}$$, there exists a Baire 1 function $$f: \mathbb{R}\to\mathbb{R}$$ such that $$\{x\in\mathbb{R}: f(x)\neq 0\}$$ is of Lebesgue measure zero and $$f+g$$ is a Darboux function for $$g\in A$$.
Reviewer: R.Pawlak (Łódź)

##### MSC:
 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A21 Classification of real functions; Baire classification of sets and functions
##### Keywords:
Baire 1 function; cliquish functions