The Automatic Baire Property and an Effective Property of [Formula: see text]-Rational Functions

We prove that [Formula: see text]-regular languages accepted by Büchi or Muller automata satisfy an effective automata-theoretic version of the Baire property. Then we use this result to obtain a new effective property of rational functions over infinite words which are realized by finite state Büchi transducers: for each such function [Formula: see text], one can construct a deterministic Büchi automaton [Formula: see text] accepting a dense [Formula: see text]-subset of [Formula: see text] such that the restriction of F to [Formula: see text] is continuous.