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Covering Baire 1 functions with darboux functions and the cofinality of the ideal of meager sets

Jordan F.

Journal of Applied Analysis

1999

Vol. 5, nr 1

71-94

Let Dar stand for the Darboux Baire class 1 functions. We show that the cofinality of the meager sets in R is the smallest cardinality of a set of Baire class 1 functions F such that for any finite collection of Baire class 1 functions G there is an f ∈ F such that f + G ⊆ Dar. Other results of this type are shown. These results are then considered as statements about additivity. The notion of super-additivity is introduced.

Some additive Darboux-like functions

Ciesielski K.

Journal of Applied Analysis

1998

Vol. 4, nr 1

43-51

In this note we will construct several additive Darboux-like functions f: R −> R answering some problems from (an earlier version of) [4]. In particular, in Section 2 we will construct, under different additional set theoretical assumptions, additive almost continuous (in sense of Stallings) functions f: R −> R whose graph is either meager or null in the plane. In Section 3 we will construct an additive almost continuous function f: R −>R which has the Cantor intermediate value property but is discontinuous on any perfect set. In particular, such an f does not have the strong Cantor intermediate value property.