  In this paper, we study the continuity of rational functions realized by
B\"uchi 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.
