On the continuity set of an Omega rational function

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 Π02-subset of Σ ω for some alphabet Σ is the continuity set C(f) of an ω-rational synchronous function f defined on Σω . 1991 Mathematics Subject Classification. 68Q05;68Q45; 03D05. LIP Research Report RR 2008-04