Continuities, Derivatives, and Differentials of $q$-Rung Orthopair Fuzzy Functions

Yager's q-rung orthopair fuzzy set (q-ROFS) is a powerful tool to handle uncertainty and vagueness in real life. It expands the spatial scope of membership and nonmembership, and therefore has a wider range of constraints and stronger modeling capabilities. However, to date, there is no investigation for q-rung orthopair fuzzy derivatives and differentials, which are very important for further developing q-rung orthopair fuzzy calculus (q-ROFC). The basic elements of a q-ROFS are q-rung orthopair fuzzy numbers (q-ROFNs), based on which we propose the q-rung orthopair fuzzy functions (q-ROFFs) and discuss their continuities in detail. Subsequently, we study the derivative of the q-ROFF, which reveals an accurate description on rate of change for continuous q-ROFF. Next, the differential operation of q-ROFF is established; thereby providing an effective approximation on nonlinear problem in the q-ROFF environment. Finally, we present numerical examples as explicit applications of q-ROFC.