Products of Świątkowski functions (Q2826352)

Let \(f\:\mathbb R\to \mathbb R\). The symbol \(\mathcal C(f)\) denotes the set of all continuity points of \(f\). We say that the function \(f\) is: {\parindent=0.7cmNEWLINE\begin{itemize} \item[--] cliquish, if the set \(\mathcal C(f)\) is dense in \(\mathbb R\);NEWLINE\item [--] Świątkowski, if for all \(a<b\) with \(f(a)\neq f(b)\), there is a \(y\) between \(f(a)\) and \(f(b)\) and an \(x\in (a,b)\cap \mathcal C(f)\) such that \(f(x)=y\). The family of all Świątkowski functions will be denoted by \(\mathcal{\acute {S}}\);NEWLINE\item [--] strong Świątkowski, if for all \(a<b\) and each \(y\) between \(f(a)\) and \(f(b)\) there is an \(x\in (a,b)\cap \mathcal C(f)\) such that \(f(x)=y\). The family of all strong Świątkowski functions will be denoted by \(\mathcal{\acute {S}}_s\).NEWLINENEWLINE\end{itemize}}NEWLINEIn this paper, the authors characterize the products of Świątkowski functions. The main result of this paper is the following:NEWLINENEWLINE Theorem. Let \(f:\mathbb R\to \mathbb R\). The following conditions are equivalent: NEWLINE{\parindent=0.7cm \begin{itemize} \item[--] there are an \(n\in \mathbb N\) and functions \(g_1,\dots,g_n\in \mathcal {\acute {S}}\) such that \(f=g_1\dots g_n\) on \(\mathbb R\),NEWLINE\item [--] \(f\) is cliquish and the set \(f^{-1}(0)\) is the union of an open set and a nowhere dense set,NEWLINE\item [--] there are functions \(g\in \mathcal{\acute {S}}_s\) and \(h\in \mathcal{\acute {S}}\) such that \(f=gh\) on \(\mathbb R\).NEWLINENEWLINE\end{itemize}}