Some continuity notions for interval functions and representation

The formalization of correctness and optimality concepts, which are desirable requirements for interval (and other self-validated methods) libraries, is related to topological aspects of real numbers and Moore arithmetics, characterizing some families of interval functions. The main motivation is that real continuous functions are largely used in many fields of human activity, and a characterization of their interval representation in terms of interval topologies leads to the knowledge of how interval algorithms are suitable to represent those functions. In this paper, we characterize and relate some classes of interval functions with respect to three natures of an interval (as a set, as an information, as a number). These natures establish different ways to classify intervals, and hence different notions of continuity. Here we relate the notion of interval representations with those classes of functions.