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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
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