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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. Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007) 0-7695-2874-0/07 $25.00

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The definition of the rough integral Pawlak proposed is improved. New notions are proposed, which are respectively the rough integral on a constant interval and the roughly integral upper limit function, and so on. By comparing with the definite integral of real functions, the properties of rough integrals are analyzed. Giving the concept of mean value of discrete functions, the mean value method for rough integration is derived. At the same time, the intermediate value theorem of rough integration is proposed and its geometric significance is analyzed, which provides a dependable theoretical tool for rough integral operation, etc.. Conclusions including the existence theorem of rough primitives and the fundamental formula of rough calculus are proposed. By the representative of discrete functions, the method of computing a primitive is given, by which basic formulas for rough integration in common use are derived, and the method of rough direct integration is obtained. There is also the method of rough integration by parts for rough integrals which is like that of definite integrals. Thus the recurrence formula for rough integration is deduced, in which the integrand of the rough integral is in the shape of the product of a rough power function and a rough exponential function. It is pointed out that integration by substitution is not applicable for rough integral operation.Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007) 0-7695-2874-0/07 $25.00

We define the dual of generalized fuzzy subspaces first. These concepts generalize the dual of fuzzy subspaces. And then we investigate the double dual of generalized fuzzy subspaces.