Almost continuous Sierpiński–Zygmund functions under different set-theoretical assumptions

Abstract

A function f:mathbb {R}rightarrow mathbb {R} is: almost continuous in the sense of Stallings, fin textrm{AC}, if each open set Gsubset mathbb {R}^2 containing the graph of f contains also the graph of a continuous function g:mathbb {R}rightarrow mathbb {R}; Sierpiński–Zygmund, fin textrm{SZ} (or, more generally, fin textrm{SZ}(textrm{Bor})), provided its restriction frestriction M is discontinuous (not Borel, respectively) for any Msubset mathbb {R} of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function f:mathbb {R}rightarrow mathbb {R} (i.e., an fin textrm{SZ}cap textrm{AC}) cannot be constructed in ZFC; however, an fin textrm{SZ}cap textrm{AC} exists under the additional set-theoretical assumption {{,textrm{cov},}}(mathcal {M})=mathfrak {c}, that is, that mathbb {R} cannot be covered by less than mathfrak {c}-many meager sets. The primary purpose of this paper is to show that the existence of an fin textrm{SZ}cap textrm{AC} is also consistent with ZFC plus the negation of {{,textrm{cov},}}(mathcal {M})=mathfrak {c}. More precisely, we show that it is consistent with ZFC+{{,textrm{cov},}}(mathcal {M})<mathfrak {c} (follows from the assumption that {{,textrm{non},}}(mathcal {N})<{{,textrm{cov},}}(mathcal {N})=mathfrak {c}) that there is an fin textrm{SZ}(textrm{Bor})cap textrm{AC} and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either {{,textrm{cov},}}(mathcal {M})=mathfrak {c} or {{,textrm{non},}}(mathcal {N})<{{,textrm{cov},}}(mathcal {N})=mathfrak {c}, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated.