{"docId":392,"paperId":392,"url":"https:\/\/dmtcs.episciences.org\/392","doi":"10.46298\/dmtcs.392","journalName":"Discrete Mathematics & Theoretical Computer Science","issn":"","eissn":"1365-8050","volume":[{"vid":82,"name":"Vol. 9 no. 2"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-00966518","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-00966518v1","dateSubmitted":"2015-03-26 16:19:23","dateAccepted":"2015-06-09 14:46:29","datePublished":"2007-01-01 08:00:00","titles":{"fr":"Baire and automata"},"authors":["Simonnet, Pierre","Cagnard, Benoit"],"abstracts":{"en":"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 \u03c9-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a B\u00fcchi 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 \u0394\u00b02, ordinals, transfinite induction and some applications of computer science."},"keywords":["[INFO.INFO-DM] Computer Science [cs]\/Discrete Mathematics [cs.DM]"]}