In this paper, we study the continuity of rational functions realized by Büchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function f has at least one point of continuity and that its continuity set C(f) cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational ${\bf \Pi}^0_2$-subset of Σω for some alphabet Σ is the continuity set C(f) of an ω-rational synchronous function f defined on Σω.

This note is about functions ƒ : Aω → Bω whose graph is recognized by a Büchi finite automaton on the product alphabet A x B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such function are continuous or not. In 1920 W. Sierpinski showed that a function $f : \mathbb{ R} \rightarrow \mathbb{R} $ is Baire class 1 if and only if both the overgraph and the undergraph of f are Fσ. We show that such characterization is also true for functions on infinite words if we replace the real ordering by the lexicographical ordering on Bω. From this we deduce that it is decidable whether such function are of Baire class 1 or not. We extend this result to real functions definable by automata in Pisot base.