Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions

Abstract:

In this paper we will investigate the smallest cardinal number $\kappa$ such that for any symmetrically continuous function $f\colon \mathbb {R}\to \mathbb {R}$ there is a partition $\{X_\xi \colon \xi <\kappa \}$ of $\mathbb {R}$ such that every restriction $f\restriction X_\xi \colon X_\xi \to \mathbb {R}$ is continuous. The similar numbers for the classes of Sierpiński-Zygmund functions and all functions from $\mathbb {R}$ to $\mathbb {R}$ are also investigated and it is proved that all these numbers are equal. We also show that $\mathrm {cf}(\mathfrak {c})\leq \kappa \leq \mathfrak {c}$ and that it is consistent with ZFC that each of these inequalities is strict.