Two notes on generalized Darboux properties and related features of additive functions

We present two results on generalized Darboux properties of additive real functions. The first results deals with a weak continuity property, called ${\bf Q}$-continuity, shared by all additive functions. We show that every ${\bf Q}$-continuous function is the uniform limit of a sequence of Darboux functions. The class of ${\bf Q}$-continuous functions includes the class of Jensen convex functions. We discuss further connections with related concepts, such as ${\bf Q}$-differentiability. Next, given a ${\bf Q}$-vector space $A\subseteq {\bf R}$ of cardinality ${\bf c}$ we consider the class ${\cal DH}^{*}(A)$ of additive functions such that for every interval $I\subseteq {\bf R}$, $f(I)=A$. We show that every function in class ${\cal DH}^{*}(A)$ can be written as the sum of a linear (additive continuous) function and an additive function with the Darboux property if and only if $A={\bf R}$. We apply this result to obtain a relativization of a certain hierarchy of real functions to the class of additive functions.