Sequential discontinuity and first-order problems

TL;DR

This paper investigates the structure of the continuous Weihrauch degrees for first-order problems, identifying a minimal discontinuous degree and analyzing sequential discontinuity without determinacy assumptions.

Contribution

It establishes the existence of a minimal discontinuous first-order degree and explores the properties of sequential discontinuity within the Weihrauch degrees framework.

Findings

Existence of a minimal discontinuous degree for first-order problems.

Identification of the least sequentially discontinuous degree.

Use of game-theoretic methods to analyze degrees.

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.