Additive Sierpiński-Zygmund functions.

Abstract

In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the class of additive Sierpiński-Zygmund (SZ) functions. In particular, we give an example of an additive Sierpiński-Zygmund (SZ) injection $f : \mathbb{R} \to\mathbb{R}$ such that $f^{-1}$ is not an SZ function. Under the assumption that $\mathbb{R}$ cannot be covered by less than $\mathfrak{c}$-many meager sets we give examples of an additive SZ bijection $f : \mathbb{R} \to\mathbb{R}$ such that $f^{-1}$ is not SZ and of an additive injection $f : \mathbb{R} \to\mathbb{R}$ such that both $f$ and $f^{-1}$ are SZ.