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

Abstract

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