Products of extra strong Świątkowski functions
For a real-valued function \(f\) defined on a non-degenerate interval \(I\), let \({\mathcal C}(f)\) denote the set of points of continuity of \(f\). Then \(f\) is called a strong Świątkowski function if for all \(\alpha,\beta\in I\) and \(y \in (f(\alpha),f(\beta))\) there is an \(x_0\in (\alpha,\beta) \cap {\mathcal C}(f)\) such that \(f(x_0)=y\). Moreover, \(f\) is called an extra strong Świątkowski function if for all \(\alpha,\beta\in I\) with \(\alpha\neq \beta\) and \(y\in [f(\alpha),f(\beta)]\) there is an \(x_0 \in [\alpha,\beta]\cap {\mathcal C}(f)\) with \(f(x_0)=y\). It is easy to see that each extra strong Świątkowski function is strong Świątkowski and each strong Świątkowski function is both Darboux and quasi-continuous. It has been proved earlier that every product of finitely many quasi-continuous Darboux functions on \(\mathbb R\) can be written as the product of just two quasi-continuous Darboux functions [\textit{A. Maliszewski}, Darboux property and quasi-continuity. A uniform approach. Słupsk: WSP (1996)], and that every product of finitely many strong Świątkowski functions on \(\mathbb R\) can be written as the product of at most four strong Świątkowski functions [\textit{P. Szczuka}, J. Appl. Anal. 12, No. 1, 129--145 (2006; Zbl 1119.26008)]. In the present paper the author proves that the latter statement is also true for extra strong Świątkowski functions (in assumption and conclusion). Moreover, a topological characterization (paralleling similar characterizations in the earlier papers) for such products is given.
scientific article; zbMATH DE number 6168742