SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS

In this paper, we study fuzzy calculus in two main branches differential and integral.  Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating  $gH$-derivative of a composite function.  Two techniques namely,  Leibniz's rule and integration by parts are introduced for fuzzy integrals.  Furthermore, we prove three essential  theorems such as a fuzzy intermediate value  theorem, fuzzy mean value theorem for integral and mean value theorem for $gH$-derivative.  We derive  a Bolzano's theorem, Rolle's theorem and some properties for $gH$-differentiable functions.  To illustrate and explain these rules and theorems, we have provided several examples in details.