Products of internally quasi-continuous functions (Q2843896)

A set \(A\subset \mathbb {R}\) is simply open if it can be written as union of an open set and a nowhere dense set. A function \(f\) is quasi-continuous if for each \(x\in \mathbb {R}\), for each open neighbourhood \(U\) of \(x\) and for each open neighbourhood \(V\) of \(f\left (x\right)\) we have int\(\left (U\cap f^{-1}\left (V\right)\right)\neq \emptyset \). A function \(f\) is internally quasi-continuous if it is quasi-continuous and its set of discontinuity points is nowhere dense. A function \(f\) is internally cliquish if the set int\(\bigl (C\left (f\right)\bigr)\) is dense in \(\mathbb {R}\), where the symbol \(C\left (f\right)\) denotes the set of all continuity points of \(f\). In the paper under review, the authors give a characterisation of the product of internally quasi-continuous functions. The main result is Theorem 4.2 saying that for each function \(f\:\mathbb {R}\rightarrow \mathbb {R}\) the following conditions are equivalent: \begin{itemize} \item\item [(i)] there are internally quasi-continuous functions \(g_1\) and \(g_2\) with \(f=g_1g_1\), \item \item [(ii)] \(f\) is a finite product of internally quasi-continuous functions, \item \item [(iii)] \(f\) is internally cliquish and the set \(\bigl \{x\in \mathbb {R}\:f\left (x\right)=0\bigr \}\) is simply open.NEWLINENEWLINEA function \(f\) is a strong Światkowski function if for each interval \(\left (a,b\right)\subset \mathbb {R}\) and for each \(\lambda \) between \(f\left (a\right)\) and \(f\left (b\right)\) there exists a point \(x_0\in \left (a,b\right)\cap C\left (f\right)\) such that \(f\left (x_0\right)=\lambda \).NEWLINENEWLINEA function \(f\) is an internally strong Światkowski function if for each interval \(\left (a,b\right)\subset \mathbb {R}\) and for each \(\lambda \) between \(f\left (a\right)\) and \(f\left (b\right)\) there exists a point \(x_0\in \left (a,b\right)\cap {\operatorname {int}}\bigl (C\left (f\right)\bigr)\) such that \(f\left (x_0\right)=\lambda \).NEWLINENEWLINEIn the paper, an example of a bounded internally quasi-continuous strong Światkowski function is constructed which cannot be written as the finite product of internally strong Światkowski functions.\end{itemize}