Continuity of functions defined on the space of fuzzy sets. (Q1429833)

The centre of discussion in this paper is \(J(\mathbb{R}^p)\) which denotes the family of all fuzzy sets \(\widetilde u:\mathbb{R}^p\to[0,1]\) with the following properties: (1) \(\widetilde u\) is normal, i.e, there exists \(x\in \mathbb{R}^p\) such that \(\widetilde u(x)=1\); (2) \(\widetilde u\) is upper semicontinuous; (3) \(\text{supp}\, \widetilde u=\overline {\{x\in \mathbb{R}^p:\widetilde u(x)>0\}}\) is compact. Addition and scalar multiplication are suitably defined and the space is endowed with a special type of metric named Skorokhod metric. The authors first deal with the question: at which points of \(I(\mathbb{R}^p)\) is a given function continuous, \(J(\mathbb{R}^p)\) being suitably topologized by the Skorokhod metric? Finally, the authors treat the continuity of the defined addition in \(J(\mathbb{R}^p)\).