Sum of Sierpiński–Zygmund and Darboux like functions

For F1,F2⊆RR we define Add(F1,F2) as the smallest cardinality of a family F⊆RR for which there is no g∈F1 such that g+F⊆F2. The main goal of this note is to investigate the function Add in the case when one of the classes F1, F2 is the class SZ of Sierpiński–Zygmund functions. In particular, we show that Martinʹs Axiom (MA) implies Add(AC,SZ)⩾ω and Add(SZ,AC)=Add(SZ,D)=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 R into R can be represented as a sum of Sierpiński–Zygmund and almost continuous functions, is independent of ZFC axioms.