We prove that ω-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 F : Σ^ω → Γ^ω , one can construct a deterministic Büchi automaton A accepting a dense Π^0_2-subset of Σ^ω such that the restriction of F to L(A) is continuous.