Neoclassical analysis: Fuzzy continuity and convergence

Abstract

The neoclassical analysis is a field in which fuzzy continuous functions are investigated. For this purpose, new measures of continuity and discontinuity (or defects of continuity) are introduced and studied. Based on such measures, classes of fuzzy continuous functions are defined and their properties are obtained. The class of fuzzy continuous functions may be considered as a fuzzy set of continuous functions. Its support consists of all functions on some topological space X into a metric space Y and its membership function is the corresponding continuity measure. Such an expansion provides a possibility to complete some important classical results. Connections between boundedness and fuzzy continuity are investigated. Criteria of boundedness and local boundedness are obtained for functions on Eucleadean spaces. Besides, such new concepts as fuzzy convergence and fuzzy uniform convergence are introduced and investigated. Their properties and connections with fuzzy continuous functions are explicated. Some results, which are obtained here, are similar to the results of classical mathematical analysis, while others differ essentially from those that are proved in classical mathematics. In the first case classical results are consequences of the corresponding results of neoclassical analysis.