Continuity in Discrete Sets.
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.
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Continuity in Discrete Sets
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