Baire and automata
Benoit Cagnard, Pierre Simonnet
Abstract
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.
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