{
  "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 $α$, let $\\mathsf{ACC}_{\\mathbb N}^α$ be the problem of solving $\\mathsf{ACC}_{\\mathbb N}$ in at most $α$ 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 $α$, and (ii) $\\mathsf{ACC}_{\\mathbb N}^α\\leq_{\\mathrm W}^{*}P$. This extends to points: $P \\geq_{\\mathrm{W}}^* \\mathsf{ACC}_{\\mathbb{N}}^α$ 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 $\\operatorname{rank}_A(p) \\leq α$ and $P$ has no continuous realizer for any neighborhood of $p$. 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 $\\mathsf{RT}^{n}_{α,β}$ and compute their ranks of discontinuity. The problems $\\mathsf{RT}^n_{α,β}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08909",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel S. Mourad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wfrz9hzeysyv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08909",
  "fulltext_url": null,
  "id": "3wfrz9hzeysyv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08909",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Computability, Logic, AI Algorithms"
    ]
  ],
  "openalex": "W7168144537",
  "pdf_url": "https://arxiv.org/pdf/2607.08909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wfrz9hzeysyv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2607.08909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08909"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Rank for Discontinuities of Multivalued Functions",
  "title_text": "A Rank for Discontinuities of Multivalued Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/3wfrz9hzeysyv",
  "venue": "arXiv",
  "versions": []
}