Decomposing Symmetrically Continuous and Sierpiński-zygmund Functions into Continuous Functions

In this paper we will investigate the smallest cardinal number κ such that for any symmetrically continuous function f : R → R there is a partition {Xξ : ξ < κ} of R such that every restriction f Xξ : Xξ → R is continuous. The similar numbers for the classes of Sierpiński-Zygmund functions and all functions from R to R are also investigated and it is proved that all these numbers are equal. We also show that cf(c) ≤ κ ≤ c and that it is consistent with ZFC that each of these inequalities is strict. 1. Preliminaries Our notation and terminology is standard and follows [5]. In particular, |X | will stand for the cardinality of X . For a cardinal number κ we will write cf(κ) for its cofinality. We also define [X ] = {Y ⊆ X : |Y | = κ}. The definition of [X ] is similar. The cardinality of the set R of real numbers is denoted by c. The functions are identified with their graphs. The class of all function from a set X into a set Y is denoted by Y X . For Z ⊂ R and a cardinal number κ ≤ c let Πκ(Z) denote the family of all coverings of Z with at most κ many sets. We will write Πκ for Πκ(R). In [4] the authors considered the following cardinal decomposition function for arbitrary families F ⊂ R , with Z ⊂ R, and G ⊂ ⋃{RX : X ⊂ Z}: dec(F ,G)= min({κ ≤ c : (∀f ∈ F)(∃X ∈ Πκ(Z))(∀X ∈ X )(f X ∈ G)} ∪ {c+}). In particular, if C stands for the family of all continuous functions from a subset of R into R, then f : R → R is countable continuous if and only if dec({f}, C) ≤ ω. In [4] the authors considered the values of dec(Bβ,Bα) for α < β < ω1, where Bα stands for the functions of α-th Baire class. In particular, they proved that cov(M) ≤ dec(B1, C) ≤ d, Received by the editors November 23, 1997 and, in revised form, February 18, 1998. 1991 Mathematics Subject Classification. Primary 26A15; Secondary 03E35.