Course overview -- Sommersemester 2012 (Archive)

Abstract: A famous question of Lusin asked whether every Borel function is countably continuous, i.e. can be written as a countable union of partial continuous functions (with arbitrary domains). This question can be straightforwardly generalized by replacing continuous functions with functions of a fixed Baire class. Both questions where answered negatively already in the Thirties, but (except for the basic case of countably continuous functions, where the Pawlikowski function provides a somewhat canonical counterexample) the desired functions are usually obtained indirectly using nontrivial Baire category argument, universal functions and diagonalization. The aim of this talk is to present new extremely simple counterexamples, which are in a sense canonical and can be seen as generalizations of the Pawlikowski function. Such examples allow also to fully describe the structure under inclusion of finite level Borel classes of functions. Finally, if time permits we will also present some (partial) positive results showing that functions appearing in certain finite level Borel classes turn out to be always countably continuous in a definable way.