Fuzzy function, Partial function, Undefined values, Functional relation

In this paper, we will study partially-defined fuzzy relations i.e., fuzzy relations with membership functions that are not necessarily defined everywhere. We will handle them in a suitable framework that is a fuzzy partial set theory. It provides tools for dealing with undefined values of the membership by means of special operations based on the connectives and quantifiers of a background fuzzy logic. We analyse a suitability of operations of a fuzzy partial set theory for a meaningful definition of functionality property. This property determines fuzzy functions and we will be concerned with their properties.