Examples of Sierpiński–Zygmund maps in the class of Darboux-like functions

Abstract

The Darboux-like functions represent a group of maps that are continuous in a generalized sense. The algebra of subsets of $${\mathbb {R}}^{\mathbb {R}}$$ (i.e., maps from $${\mathbb {R}}$$ to $${\mathbb {R}}$$) generated by these classes has nine atoms, that is, the smallest non-empty elements of the algebra. The subject of this work is to study the intersections of these atoms with the class $${{\,\mathrm{SZ}\,}}$$ of Sierpinski–Zygmund functions—the maps that have as little of the standard continuity as possible. Specifically, we will show that it is independent of the standard axioms of set theory that each of these atoms has a non-empty intersection with $${{\,\mathrm{SZ}\,}}$$. For seven of the nine atoms this has been unknown, and the constructions of the examples provide answers to the problems stated in a recent survey A century of Sierpinski–Zygmund functions of K. C. Ciesielski and J. Seoane-Sepulveda. Notice that lineability of the main classes of Darboux-like functions, as well as of Sierpinski–Zygmund functions, has been intensively studied. The presented work opens a possibility to study also the lineability of the nine smaller classes we discuss here.