On additive almost continuous functions under CPA game prism

Abstract. We prove that the Covering Property Axiom CPAgameprism, which holds in the iterated perfect set model, implies that there ex-ists an additive discontinuous almost continuous function f: R → R whose graph is of measure zero. We also show that, under CPAgameprism, there exists a Hamel basis H for which, E+(H), the set of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin’s axiom, while it is unknown whether either of them can be constructed in ZFC. As a tool for the constructions we will show that CPAgameprism implies its seemingly stronger version, in which ω1-many games are played si-multaneously