Sequential discontinuity and first-order problems
Arno Pauly, Giovanni Soldà
TL;DR
The paper addresses identifying the minimal nontrivial first-order part in the continuous Weihrauch lattice and understanding how discontinuity manifests for first-order multivalued problems. It develops a framework based on generalized Wadge games to connect discontinuity with the degree $ACC_{\mathbb{N}}$, using domain $\omega+1$ to construct a representative $\mathalpha{\neq}_{\mathbf{X}}$ and establish key equivalences between first-order discontinuity, convergent sequences in the domain, and reductions to $ACC_{\mathbb{N}}$. The authors prove that $ACC_{\mathbb{N}}$ is the least nontrivial first-order degree, derive dichotomies between continuity and $\Pi^0_2\mathsf{ACC}_{\mathbb{N}}$-type lower bounds via backtrack-like games, and relate these results to admissibility of represented spaces. The work clarifies the low-level structure of continuous Weihrauch degrees, links sequential discontinuity to first-order parts, and provides a foundation for further domain classifications and dichotomy results in computable analysis.
Abstract
We explore the low levels of the structure of the continuous Weihrauch degrees of first-order problems. In particular, we show that there exists a minimal discontinuous first-order degree, namely that of $\accn$, without any determinacy assumptions. The same degree is also revealed as the least sequentially discontinuous one, i.e. the least degree with a representative whose restriction to some sequence converging to a limit point is still discontinuous. The study of games related to continuous Weihrauch reducibility constitutes an important ingredient in the proof of the main theorem. We present some initial additional results about the degrees of first-order problems that can be obtained using this approach.