On the sum of functions with condition s3.

Abstract

A function $f:{\mathR} \to {\mathR}$ satisfies condition $(s_3)$ if for each real $\varepsilon > 0$, for each $x$ and for each set $U \ni x$ belonging to the density topology there is an open interval $I$ such that $A(f) \supset I \cap U \neq \emptyset $ and $f(U\cap I) \subset (f(x)-\varepsilon ,f(x)+\varepsilon )$, where $A(f)$ denotes the set of all approximate continuity points of $f$. In this article it is show that the sum of two functions with the condition $(s_3)$ is the sum of two Darboux functions satisfying this condition $(s_3)$ and that every a.e.-continuous function with some special condition is the sum of two functions with condition $(s_3).$