Diagonals of separately continuous multi-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.