Continuity in Discrete Sets - Mathematics > Classical Analysis and ODEsReport as inadecuate

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Abstract: Continuous models used in physics and other areas of mathematics applicationsbecome discrete when they are computerized, e.g., utilized for computations.Besides, computers are controlling processes in discrete spaces, such as filmsand television programs. At the same time, continuous models that are in thebackground of discrete representations use mathematical technology developedfor continuous media. The most important example of such a technology iscalculus, which is so useful in physics and other sciences. The main goal ofthis paper is to synthesize continuous features and powerful technology of theclassical calculus with the discrete approach of numerical mathematics andcomputational physics. To do this, we further develop the theory of fuzzycontinuous functions and apply this theory to functions defined on discretesets. The main interest is the classical Intermediate Value theorem. Althoughthe result of this theorem is completely based on continuity, utilization of arelaxed version of continuity called fuzzy continuity, allows us to provediscrete versions of the Intermediate Value theorem. This result providesfoundations for a new approach to discrete dynamics.

Author: Mark Burgin


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