Baire and automata

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science