Universally Bad Darboux Functions in the Class of Additive Functions

Banaszewski, Dariusz

doi:10.2307/44152751

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2018

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Cited by 1 publication

(2 citation statements)

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“…We provide below examples of "universally bad" functions in the class of additive functions, in a manner similar in spirit to results in [2], [18], [34]:…”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

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

Istrate

2018

Preprint

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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.

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“…We provide below examples of "universally bad" functions in the class of additive functions, in a manner similar in spirit to results in [2], [18], [34]:…”

Section: Proofmentioning

confidence: 99%

“…Therefore it is often believed that such functions cannot have any nice property." But this is not the case: since the writing of [17], a Ph.D. thesis [3] and several papers (among them [1,2,9,10,11,33,35]) have investigated Darboux-like properties and relaxations of continuity for additive functions.…”

Section: Introductionmentioning

confidence: 99%