Decomposing Baire class one functions (Q2874884)

Recall that a function \(f:\mathcal{N}\to \mathcal{N}\) is said to be \textit{Baire class one} if the inverse image of any open set is \(\Sigma_2^0\). (Here, \(\mathcal{N}\) denotes the Baire space \(\omega^\omega\).) Recall also Baire's classical theorem, namely that a function on Baire space is Baire class one if and only if, for any closed subset \(A\) of Baire space, the points of \(A\) that are continuity points for the function are dense in \(A\).NEWLINENEWLINEThis paper reviews the study of Baire class one functions using certain games and certain ordinal ranks on functions on the Baire space. In particular, the author introduces the \textit{discontinuity} of a function \(f:\mathcal{N}\to \mathcal{N}\), denoted \(\text{disc}(f)\), to be the rank associated to the Borel derivative on subsets of Baire space given by sending a closed set \(D\) to the closure of the set of points of \(D\) that are points of discontinuity of \(f\). The article establishes some basic properties of these games and ranks and concludes with the following theorem: Suppose that \(f:\mathcal{N}\to \mathcal{N}\) is Baire class one. Then, the inverse image of some open set is \(\Sigma_2^0\)-complete if and only if \(\text{disc}(f)=\omega_1\) if and only if there is a perfect set \(P\) such that the discontinuity points of \(f|P\) are dense in \(P\). Also, the inverse image of every open set is \(\Delta_2^0\) if and only \(\text{disc}(f)<\omega_1\) if and only if for any perfect set \(P\), the continuity points of \(f|P\) contain a dense open subset of \(P\).NEWLINENEWLINEFor the entire collection see [Zbl 1280.03005].