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. 6176, 2015.
Abstract:
We present two results on generalized Darboux properties of additive real functions.
The first one introduces a weak continuity property, called Qcontinuity, possessed by all
additive functions. We show that every
Qcontinuous function is the uniform limit of a sequence of Darboux functions. The class of Qcontinuous functions includes the class of Jensen convex functions. We discuss further connections with related concepts, such as Qdifferentiability.
Next, given Qvector 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
URL:
http://tcs.ieat.ro/wpcontent/uploads/2015/02/marcushamel.pdf
Posted by
Gabriel Istrate
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