On the Darboux property of the sum of cliquish functions (Q1194659)

The starting point of the considerations of this paper is the well-known theorem proved by \textit{H. W. Pu} and \textit{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 \(g\in A\).