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Krzysztof, Chris and Natkaniec, T. and Rodríguez-Vidanes, D.L.
*Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions.*
(Unpublished)

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## Abstract

A 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.

Item Type: | Article |
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Uncontrolled Keywords: | Additivity Almost continuous functions; Covering of category; Covering of measure; Lineability; Random reals; Sierpiński-Zygmund functions |

Palabras clave (otros idiomas): | Espacios vectoriales |

Subjects: | Sciences > Mathematics Sciences > Mathematics > Mathematical analysis |

ID Code: | 73501 |

Deposited On: | 07 Jul 2022 11:28 |

Last Modified: | 07 Jul 2022 12:36 |

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