The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function

TL;DR

This paper investigates the descriptive set-theoretic complexity of the set of points where a multi-valued function is continuous, establishing conditions for the set to be Borel or a union of $G_\delta$ sets, and providing counterexamples.

Contribution

It characterizes the set-theoretic complexity of points of continuity for multi-valued functions, including necessary conditions and counterexamples, extending understanding of their descriptive complexity.

Findings

Conditions for the set of continuity points to be $G_\delta$ or a countable union of $G_\delta$ sets.

Counterexamples showing the optimality of these conditions.

Results on stronger notions of continuity for multi-valued functions.

Abstract

In this article we treat a notion of continuity for a multi-valued function $F$ and we compute the descriptive set-theoretic complexity of the set of all $x$ for which $F$ is continuous at $x$. We give conditions under which the latter set is either a $G_\delta$ set or the countable union of $G_\delta$ sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of $F$ is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function $F$ whose graph is a Borel set and the set of points of continuity of $F$ is not a Borel set. Finally we give some analogous results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in [{\em Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra}, {\sl submitted}].