On the continuity set of an Omega rational function

Carton, Olivier; Finkel, Olivier; Simonnet, Pierre

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 183-196.

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 $C\left(f\right)$ 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 ${\Pi }_2^0$-subset of ${\Sigma }^{\omega }$ for some alphabet $\Sigma$ is the continuity set $C\left(f\right)$ of an $\omega$-rational synchronous function $f$ defined on ${\Sigma }^{\omega }$.

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