For classes F_{1} and F_{2}
of functions from **R** into **R**
we define
Add(F_{1},F_{2}) as the smallest cardinality of a family
F of sunctions
from X
into **R** for which there is no g in F_{1} such that
g+F is a subset of F_{2}.
The main goal of this note is to investigate the function
Add in the case when one of the classes F_{1}, F_{2} 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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**Last modified June 29, 2001.**