On the issue of discrete smoothness

Abstract

Discrete Mathematical Analysis (DMA) is a new approach to data analysis, focused on the
researcher and occupying an intermediate position between hard mathematical methods and soft fuzzy methods.
Fuzzy Sets (FS) play an important role in DMA, some of which are models of discrete
analogs of fundamental mathematical properties (proximity, limit, trend, connectivity, . . .), as well as Fuzzy Logic (FL), which allows combining fuzzy models into data analysis algorithms, in particular, according to classical mathematical scenarios.
In DMA, a regression approach to the limit and derivative is adopted: they are, respectively,
the value and slope of a linear regression, constructed based on a function and fuzzy structure on the initial finite space, modeling the limit transition at its point.
Thus, the regression limit and regression derivative always exist. The question arises about
their quality, in particular, the ability to detect discrete smoothness. This requires a more
in-depth regression analysis than traditional methods, which is the focus of this paper.