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 functionsand 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 theovergraph and the undergraph of f are Fσ . We show thatsuch characterization is also true for functions on infinite wordsif we replace the real ordering by the lexicographical orderingon Bω . From this we deduce that it is decidable whethersuch function are of Baire class 1 or not. We extend this resultto real functions definable by automata in Pisot base.