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.

中文翻译：

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类 Darboux 函数类中的 Sierpiński-Zygmund 映射示例

类 Darboux 函数表示一组广义上连续的映射。$${\mathbb {R}}^{\mathbb {R}}$$ 子集的代数（即从 $${\mathbb {R}}$$ 映射到 $${\mathbb {R}}这些类生成的 $$) 有九个原子，即代数中最小的非空元素。这项工作的主题是研究这些原子与类 $${{\,\mathrm{SZ}\,}}$$ 的 Sierpinski-Zygmund 函数的交集——这些映射的标准连续性与可能的。具体来说，我们将证明，这些原子中的每一个都与 $${{\,\mathrm{SZ}\,}}$$ 有一个非空交集，这与集合论的标准公理无关。对于九个原子中的七个，这是未知的，示例的构造为最近的调查中陈述的问题提供了答案 K. C. Ciesielski 和 J. Seoane-Sepulveda 的谢尔宾斯基-齐格蒙德函数的一个世纪。请注意，已经深入研究了类 Darboux 函数以及 Sierpinski-Zygmund 函数的主要类别的线性。所呈现的工作为研究我们在这里讨论的九个较小类的可直线性提供了可能性。