  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.
