  It was noticed by Harel in [Har86] that "one can define $\Sigma_1^1$-complete
versions of the well-known Post Correspondence Problem". We first give a
complete proof of this result, showing that the infinite Post Correspondence
Problem in a regular $\omega$-language is $\Sigma_1^1$-complete, hence located
beyond the arithmetical hierarchy and highly undecidable. We infer from this
result that it is $\Pi_1^1$-complete to determine whether two given infinitary
rational relations are disjoint. Then we prove that there is an amazing gap
between two decision problems about $\omega$-rational functions realized by
finite state B\"uchi transducers. Indeed Prieur proved in [Pri01, Pri02] that
it is decidable whether a given $\omega$-rational function is continuous, while
we show here that it is $\Sigma_1^1$-complete to determine whether a given
$\omega$-rational function has at least one point of continuity. Next we prove
that it is $\Pi_1^1$-complete to determine whether the continuity set of a
given $\omega$-rational function is $\omega$-regular. This gives the exact
complexity of two problems which were shown to be undecidable in [CFS08].
