The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (Extended Abstract)

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 its classification as G_delta or Borel, and providing counterexamples.

Contribution

It characterizes the complexity of the continuity set for multi-valued functions and identifies necessary conditions for Borel measurability, extending previous understanding in descriptive set theory.

Findings

Conditions for the continuity set to be G_delta or a union of G_delta sets.

Counterexamples showing optimality of these conditions.

Necessary conditions for the continuity set to be Borel.

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 analogue results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra", (submitted).