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Sum of Sierpinski-Zygmund and Darboux Like Functions

Krzysztof Plotka

Abstract 
For classes F1 and F2 of functions from R into R we define Add(F1,F2) as the smallest cardinality of a family F of sunctions from X into R for which there is no g in F1 such that g+F is a subset of 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 Sierpinski-Zygmund functions. In particular, we show that Martin's Axiom (MA) implies Add(AC,SZ) >= \omega and $Add(SZ,AC)= Add(SZ,D) = \continuum, 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 Sierpinski-Zygmund and almost continuous functions is independent of ZFC axioms.

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