Sum of Sierpiński–Zygmund and Darboux like functions

Abstract

For $\text{F}{}_1\text{,}\text{F}{}_2\text{⊆}\text{R}{}^{\mathrm{<mtext>R</mtext>}}$ we define $\text{Add}\text{(}\text{F}{}_1\text{,}\text{F}{}_2\text{)}$ as the smallest cardinality of a family $\text{F⊆}\text{R}{}^{\mathrm{<mtext>R</mtext>}}$ for which there is no $\text{g∈}\text{F}{}_1$ such that $\text{g+F⊆}\text{F}{}_2$. The main goal of this note is to investigate the function Add in the case when one of the classes $\text{F}{}_1$, $\text{F}{}_2$ is the class SZ of Sierpiński–Zygmund functions. In particular, we show that Martin's Axiom (MA) implies Add(AC,SZ)⩾ω and $\text{Add}\text{(}\text{SZ}\text{,}\text{AC}\text{)=}\text{Add}\text{(}\text{SZ}\text{,}\text{D}\text{)=}\text{c}$, where AC and D denote the families of almost continuous and Darboux functions, respectively. As a corollary we obtain that the proposition, every function from $\text{R}$ into $\text{R}$ can be represented as a sum of Sierpiński–Zygmund and almost continuous functions, is independent of ZFC axioms.