Two Effective Properties of ω-Rational Functions

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, 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 [Formula: see text] to [Formula: see text] is continuous. Secondly, we give a new proof of the decidability of the first Baire class for synchronous [Formula: see text]-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular [Formula: see text]-languages.