Continuity in Discrete Sets

TL;DR

This paper develops a theory of fuzzy continuous functions to bridge continuous calculus and discrete mathematics, enabling discrete versions of the Intermediate Value theorem for applications in computational physics and discrete dynamics.

Contribution

It introduces fuzzy continuity to extend classical calculus results to discrete sets, creating a foundation for new discrete dynamical systems analysis.

Findings

Fuzzy continuity generalizes classical continuity for discrete sets.

Discrete Intermediate Value theorem is proven using fuzzy continuity.

Lays groundwork for new approaches in discrete dynamics.

Abstract

Continuous models used in physics and other areas of mathematics applications become discrete when they are computerized, e.g., utilized for computations. Besides, computers are controlling processes in discrete spaces, such as films and television programs. At the same time, continuous models that are in the background of discrete representations use mathematical technology developed for continuous media. The most important example of such a technology is calculus, which is so useful in physics and other sciences. The main goal of this paper is to synthesize continuous features and powerful technology of the classical calculus with the discrete approach of numerical mathematics and computational physics. To do this, we further develop the theory of fuzzy continuous functions and apply this theory to functions defined on discrete sets. The main interest is the classical Intermediate Value theorem. Although the result of this theorem is completely based on continuity, utilization of a relaxed version of continuity called fuzzy continuity, allows us to prove discrete versions of the Intermediate Value theorem. This result provides foundations for a new approach to discrete dynamics.