Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions

dc.contributor.authorKrzysztof, Chris
dc.contributor.authorNatkaniec, T.
dc.contributor.authorRodríguez-Vidanes, D.L.
dc.descriptionCRUE-CSIC (Acuerdos Transformativos 2022)
dc.description.abstractA function f : R → R is: almost continuous in the sense of Stallings, f ∈ AC, if each open set G ⊂ R2 containing the graph of f contains also the graph of a continuous function g : R → R; Sierpiński-Zygmund, f ∈ SZ (or, more generally, f ∈ SZ(Bor)), provided its restriction f M is discontinuous (not Borel, respectively) for any M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński-Zygmund almost continuous function f : R → R (i.e., an f ∈ SZ ∩ AC) cannot be constructed in ZFC; however, an f ∈ SZ ∩ AC exists under the additional set-theoretical assumption cov(M) = c, that is, that R cannot be covered by less than c-many meager sets. The primary purpose of this paper is to show that the existence of an f ∈ SZ∩AC is also consistent with ZFC plus the negation of cov(M) = c. More precisely, we show that it is consistent with ZFC+cov(M) < c (follows from the assumption that non(N ) < cov(N ) = c) that there is an f ∈ SZ(Bor)∩AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either cov(M) = c or non(N ) < cov(N ) = c, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński-Zygmund functions. Several open problems are also stated.
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.sponsorshipMinisterio de Ciencia e Innovación (MICINN)/FEDER
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dc.journal.titleRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas volume
dc.publisherSpringer Nature
dc.relation.projectIDPGC2018-097286-B-I00; PRE2019-089135
dc.rightsAtribución 3.0 España
dc.rights.accessRightsopen access
dc.subject.keywordAdditivity Almost continuous functions
dc.subject.keywordCovering of category
dc.subject.keywordCovering of measure
dc.subject.keywordRandom reals
dc.subject.keywordSierpiński-Zygmund functions
dc.subject.keywordEspacios vectoriales
dc.subject.ucmMatemáticas (Matemáticas)
dc.subject.ucmAnálisis matemático
dc.subject.unesco12 Matemáticas
dc.subject.unesco1202 Análisis y Análisis Funcional
dc.titleAlmost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions
dc.typejournal article
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