Sierpiński-Zygmund uniform limits of extendable connectivity functions. (Q595850)

\(f:{\mathbb R}\to{\mathbb R}\) is: a Sierpiński-Zygmund (SZ) function if \(f| A\) is continuous for no subset \(A\) of \(\mathbb R\) of cardinality of the continuum; an extendable (Ext) function if there is a function \(F:{\mathbb R}\times [0,1]\to{\mathbb R}\) such that \(F(x,0)=f(x)\) for all \(x\in{\mathbb R}\) and \(F| C\) is connected for each connected subset \(C\subset{\mathbb R}\times [0,1]\). It is known that Ext\(\cap \text{SZ}=\emptyset\) [see e.g.,\textit{K. Banaszewski} and \textit{T. Natkaniec}, Real Anal. Exch. 24, No. 2, 827--835 (1998; Zbl 0967.26002)]. The first part of the paper under review contains the construction of a function \(f\colon{\mathbb R}\to{\mathbb R}\) which is the uniform limit of a sequence of extendable functions (i.e., \(f\in\overline{\text{Ext}}\)). In the second part of the paper it is shown under MA (in fact, under the assumption that every set \(A\subset\mathbb R\) of size less than the continuum is meager) that the class SZ\(\cap \overline{\text{Ext}}\) cannot be characterized by preimages of sets, i.e., there are no \({\mathcal A},{\mathcal B}\subset{\mathcal P}(\mathbb R)\) such that SZ\(\cap \overline{ \text{Ext}}\) is equal to the class of all \(f\colon{\mathbb R}\to{\mathbb R}\) with \(f^{-1}(B)\in{\mathcal A}\) for each \(B\in {\mathcal B}\). Finally the author poses the question whether or not the class SZ\(\cap \overline{\text{Ext}}\) can be characterized by images of sets.