Mazurkiewicz sets and containment of Sierpiński–Zygmund functions under rotations

A Mazurkiewicz set is a plane subset that intersects every straight line at exactly two points, and a Sierpiński–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 Sierpiński–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 Sierpiński–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.