A Rank for 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 α, let ACCℕα be the problem of solving ACCℕ 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) ACCℕα≤W*P. This extends to points: P ≥W* ACCℕα via a forward function that sends #ℕ to p ∈ dom(P) if and only if P is discontinuous on a set A with rankA(p) ≤ 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 RTnk,j to ordinal parameters, we define RTnα,β and compute their ranks of discontinuity. The problems RTnα,β provide examples of problems with discontinuities of each countable rank which are not reducible to ACCℕ. The separation from ACCℕ is obtained via the notion of guessability with identified errors.
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BibTeX
@misc{mourad2026rank,
title = {{A Rank for Discontinuities of Multivalued Functions}},
author = {Daniel S. Mourad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08909},
eprint = {2607.08909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08909},
}
RIS
TY - UNPB
TI - A Rank for Discontinuities of Multivalued Functions
AU - Daniel S. Mourad
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2607.08909
UR - https://arxiv.org/abs/2607.08909
AB - 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 ACC_ℕ^α be the problem of solving ACC_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) ACC_ℕ^α≤_W^*P. This extends to points: P ≥_W^* ACC_ℕ^α via a forward function that sends #^ℕ to p ∈ dom(P) if and only if P is discontinuous on a set A with rank_A(p) ≤ 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 RT^n_k,j to ordinal parameters, we define RT^n_α,β and compute their ranks of discontinuity. The problems RT^n_α,β provide examples of problems with discontinuities of each countable rank which are not reducible to ACC_ℕ. The separation from ACC_ℕ is obtained via the notion of guessability with identified errors.
ER -