The continuity of 𝑝-rationality of characters and the principal block

We study rationality properties of irreducible characters of finite groups. We show that the continuity of 2-rationality is a phenomenon that can be detected in the principal 2-block, thus refining a recent result of N. N. Hung. We also propose a conjectural group theoretical criterion for the continuity gap at level 1 for all primes.