On additive functions with additional derivation properties

The purpose of this paper is to introduce the notion a generalized derivation which derivates prescribed family smooth vector-valued functions several variables. basic calculus rules are established and then result derived shows that if function $f$ satisfies an addition theorem whose determining operation derivable with respect additive $d$, itself $d$. As application approach, new proof generalization recent Maksa obtained. We also extend Nishiyama Horinouchi formulate two open problems.