Mazurkiewicz Sets and Containment of Sierpi\'{n}ski-Zygmund Functions under Rotations

Abstract

A Mazurkiewicz set is a plane subset that intersect every straight line at exactly two points, and a Sierpi\'{n}ski-Zygmund function is a function from $\mathbb{R}$ into $\mathbb{R}$ that has as little of the standard continuity as possible. Building on the recent work of Kharazishvili, we construct a Mazurkiewicz set that contains a Sierpi\'{n}ski-Zygmund function in every direction and another one that contains none in any direction. Furthermore, we show that whether a Mazurkiewicz set can be expressed as a union of two Sierpi\'{n}ski-Zygmund functions is independent of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Some open problems related to the containment of Hamel functions are stated.