Diagonals of separately continuous multi-valued mappings

TL;DR

This paper constructs examples of separately continuous multi-valued mappings with specific diagonal properties, revealing limitations in extending joint continuity results from compact-valued to closed-valued mappings.

Contribution

It provides a method to construct separately continuous mappings with prescribed diagonals and presents a counterexample showing the limits of existing continuity theorems.

Findings

Constructed a separately continuous mapping with a given diagonal as a pointwise limit.

Provided a counterexample of a closed-valued separately continuous mapping with an everywhere discontinuous diagonal.

Showed that joint continuity results for compact-valued mappings do not extend to closed-valued mappings.

Abstract

We solve a problem on a construction of a separately continuous mapping with the given diagonal, which is the pointwise limit of a sequence of continuous mappings valued in an equiconnected space. We construct an example of a closed-valued separately continuous mapping $f:[0,1]^2\to \mathbb R$ with an everywhere discontinuous diagonal. The example shows that the results on the joint continuity point set of compact-valued separately continuous mappings can not be generalized to the case of closed-valued mappings.