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. We also give a new proof of the decidability of the first Baire class for synchronous $\omega$-rational functions from which we get an extension of this result.