Discrete Mathematics & Theoretical Computer Science,Vol 9, No 2 (2007)

摘要

In his thesis Baire defined functions of Baire class 1. A functionf is of Baire class 1 if it is the pointwise limit of asequence of continuous functions. Baire proves the followingtheorem. A function f is not of class 1 if and only ifthere exists a closed nonempty set F such that therestriction of f to F has no point ofcontinuity. We prove the automaton version of this theorem. Anω-rational function is not of class 1 if and onlyif there exists a closed nonempty set F recognized by aBüchi automaton such that the restriction of f toF has no point of continuity. This gives us theopportunity for a discussion on Hausdorff's analysis ofΔ°2, ordinals, transfiniteinduction and some applications of computer science.