Ranked Discontinuities of Multivalued Functions

Abstract

We study discontinuity of multivalued functions (also known as problems) $P$ on Baire space by assigning an ordinal rank to points in the domain of $P$ that have no local realizers. For each countable ordinal $\alpha$, let $\mathsf{ACC}_{\mathbb N}^{\alpha}$ be the problem of solving $\mathsf{ACC}_{\mathbb N}$ in at most $\alpha$ many attempts, with a new instance provided for each attempt. Our main theorem shows that, for any problem $P$, the following are equivalent: (i) $P$ is discontinuous on some set all of whose points have Cantor--Bendixson rank at most $\alpha$, and (ii) $\mathsf{ACC}_{\mathbb N}^{\alpha}\leq_{\mathrm W}^{*}P$. This extends to points: $P \geq_{\mathrm{W}}^* \mathsf{ACC}_{\mathbb{N}}^\alpha$ via a forward function that sends $\#^{\mathbb{N}}$ to $p \in \operatorname{dom}(P)$ if and only if $P$ is discontinuous on a set $A$ with $\operatorname{rank}_A(p) \leq \alpha$. We also characterize these properties via a Wadge-style discontinuity game for $P$. We apply this framework to the thin set and achromatic Ramsey theorems. Extending $\mathsf{RT}^{n}_{k,j}$ to ordinal parameters, we define $\mathsf{RT}^{n}_{\alpha,\beta}$ and compute their ranks of discontinuity. The problems $\mathsf{RT}^n_{\alpha,\beta}$ provide examples of problems with discontinuities of each countable rank which are not reducible to $\mathsf{ACC}_{\mathbb{N}}$. The separation from $\mathsf{ACC}_{\mathbb{N}}$ is obtained via the notion of guessability with identified errors.