  We prove that $\omega$-regular languages accepted by B\"uchi 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 B\"uchi
transducers: for each such function $F: \Sigma^\omega \rightarrow
\Gamma^\omega$, one can construct a deterministic B\"uchi automaton
$\mathcal{A}$ accepting a dense ${\bf \Pi}^0_2$-subset of $\Sigma^\omega$ such
that the restriction of $F$ to $L(\mathcal{A})$ is continuous.
