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Gabriel Istrate. Two notes on generalized Darboux properties and related features of additive functions. Analele Universitatii Bucuresti Ser. Informatica, special issue dedicated to Professor Solomon Marcus' s 90th birthday , LXII (2), pp. 61-76, 2015.

Abstract: We present two results on generalized Darboux properties of additive real functions.

The first one introduces a weak continuity property, called Q-continuity, possessed 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 Q-vector space $A subseteq R$ of cardinality c, we consider the class DH^{*}(A) of additive functions such that for every interval $Isubseteq R$,
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.

Keywords: Darboux properties, additive function, real analysis


Posted by Gabriel Istrate


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