An Improved Intermediate Value Theorem and Rough Fix-Point Theorem of Roughly Continuous Discrete Functions

The incompleteness and insufficiency in the investigation of rough continuity in Pawlak rough function model are pointed out. Proposing the e-d definition of rough continuity of a discrete function, the concept of Pawlak rough continuity is deduced in the form of a proposition. A series of operating properties of roughly continuous functions are discussed such as maximization, minimization, complementarity, etc.. By an opposite example, we point out that the sufficient condition in the intermediate value theorem Pawlak proposed is not valid. An improved intermediate value theorem of roughly continuous discrete functions on closed intervals is then proposed. The concept of connectivity function closely connected with the rough continuity is introduced, by which the improved intermediate value theorem is proved strictly. Moreover, the concept of rough fix-points of discrete functions is introduced. The rough fix-point theorem of roughly continuous functions is proposed and studied, and some new results are achieved.