Different notions of Sierpiński–Zygmund functions

Abstract

A function $f:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ is Sierpiński–Zygmund, $f\in {{\,\mathrm{SZ}\,}}(\mathrm {C})$, provided its restriction $f{\restriction }M$ is discontinuous for any $M\subset {{\mathbb {R}}}$ of cardinality continuum. Often, it is slightly easier to construct a function $f:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$, denoted as $f\in {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$, with a seemingly stronger property that $f{\restriction }M$ is not Borel for any $M\subset {{\mathbb {R}}}$ of cardinality continuum. It has been recently noticed that the properness of the inclusion ${{\,\mathrm{SZ}\,}}(\mathrm {Bor})\subseteq {{\,\mathrm{SZ}\,}}(\mathrm {C})$ is independent of ZFC. In this paper we explore the classes ${{\,\mathrm{SZ}\,}}(\Phi )$ for arbitrary families $\Phi$ of partial functions from ${{\mathbb {R}}}$ to ${{\mathbb {R}}}$. We investigate additivity and lineability coefficients of the class ${{\mathbb {S}}}:={{\,\mathrm{SZ}\,}}(\mathrm {C}){\setminus } {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$. In particular we show that if ${{\mathfrak {c}}}=\kappa ^+$ and ${{\mathbb {S}}}\ne \emptyset$, then the additivity of ${{\mathbb {S}}}$ is $\kappa$, that ${{\mathbb {S}}}$ is ${{\mathfrak {c}}}^+$-lineable, and it is consistent with ZFC that ${{\mathbb {S}}}$ is ${{\mathfrak {c}}}^{++}$-lineable. We also construct several examples of functions from ${{\,\mathrm{SZ}\,}}(\mathrm {C}){\setminus } {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$ that belong also to other important classes of real functions.