Rough Derivatives in Rough Function Model and Their Application

On the theory and application of rough derivatives in rough function model, the notion of generalized rough functions is defined to improve the definition of higher order rough derivatives Pawlak proposed, and analyze the functional features of roughly derived functions and higher order roughly derived functions. The rough extremum definition of discrete functions is given, Fermat theorem and Rolle theorem of roughly smooth discrete functions are proposed and proved, which complete and perfect the theoretical foundation of rough derivatives application in rough function model. The concepts that are rough monotone and rough convexity of discrete functions are defined. By comparing with the derivative application of real continuous function, a series of theorems are put forward which are the relation theorem of rough derivatives and rough monotone, the two sufficient conditions of rough extrema, the two relation theorems of rough derivatives and rough convexity. Moreover, some new results are achieved such as the sufficient condition of rough smoothing of discrete functions, etc.