In the 1920s, Lusin asked whether every Borel function on a Polish space is a union of countably many partial continuous functions (i.e. whether every Borel function is σ-continuous). This question has a negative answer; an example of a non-piecewise continuous Borel function is the Turing jump. A dichotomy of Solecki and Zapletal is that the Turing jump is the basis for every counterexample: every Borel function f is either σ-continuous, or the Turing jump continuously reduces to f.
We generalize the Solecki-Zapletal dichotomy throughout the Borel hierarchy. Recall that a Borel function is Baire class α if and only if it is Σ0α+1-measurable. We show that every Borel function f is either σ-Baire class α, or a complete Baire class α+1 function (an appropriate iterate of the Turing jump) continuously reduces to f. Our proof uses an adaptation of Montalbán's game metatheorem for priority arguments to boldface descriptive set theory.
This is joint work with Antonio Montalbán.