2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146619In 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 Pi^0_2-subset of X^omega for some alphabet X is the continuity set C(F) of an omega-rational synchronous function F defined on X^omega.Dedicated to Serge Grigorieff on the occasion of his 60th BirthdayComputational ComplexityLogic in Computer Sciencetext