More about Sierpiński-Zygmund uniform limits of extendable functions

Abstract

Let $SZ$, $D$, $Ext$, and $\overline{Ext}$ denote respectively the spaces of Sierpiński-Zygmund functions, Darboux functions, extendable connectivity functions, and uniform limits of sequences of extendable connectivity functions, with the metric of uniform convergence on them. We show that the subspaces $SZ \cap D$ and $SZ \cap \overline{Ext}$ are each porous in the space $SZ$, but $SZ \cap \overline{Ext}$ is not porous in the space $\overline{Ext}$. We also show that every real function can be expressed as a sum of two Sierpiński-Zygmund functions one of which belongs to $\overline{Ext}$. Ciesielski and Natkaniec showed in 1997 that if $\R$ is not the union of less than $\mathfrak c$-many nowhere dense subsets, then there exist Sierpiński-Zygmund bijections $f,g:\mathbb{R}\to\mathbb{R}$ such that $f^{-1} \notin SZ$ and $g^{-1}\in SZ$, but here we can additionally have $f$ and $g$ belonging to $\overline{Ext}$