### 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.Full Text: PDF PostScript