Authors: Pierre Simonnet ^{1}; Benoit Cagnard ^{1}
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Pierre Simonnet;Benoit Cagnard
1 Sciences pour l'environnement
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.