A Modern Introduction to Fuzzy Mathematics

12Fuzzy Calculus

The importance of fuzzy calculus becomes strongly evident if one thinks of the role played by classical calculus concepts in the mathematical modeling of real‐world phenomena. There, the complexity of nature forces one to make a lot of compromise, that is, to formulate assumptions, assertions, and premises not only in order to build and apply a mathematically tractable model, but also because of the imprecision or vagueness often characterizing the gathered experimental information or even the theoretical background needed to quantitatively describe a particular natural phenomenon. A possible way to deal with the aforesaid vagueness in mathematical modeling is offered by fuzzy calculus. In this chapter, we shall first define fuzzy functions and their properties and then examine their integration and differentiation. Thereafter, we consider a theory of fuzzy limits, and finally, we shall study fuzzy differential equations.

12.1 Fuzzy Functions

The possibility of fuzziness for a real‐valued function of one independent real variable, which henceforth will be called fuzzy function, can rather naturally appear basically in three different ways (see, e.g., [190] or [110]):

1. the nonfuzzy function considered has fuzzy constraints on its domain and codomain;
2. there is fuzziness in the independent variable, and the nonfuzzy function “transfers” this fuzziness to the codomain but without producing new fuzziness; and
3. a nonfuzzy function is properly extended so that it ...