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

Notes for rough derivatives and rough continuity in rough function model

“…[14] only proved the former. In the following, two mappings are defined to verify the higher order rough derivative formula.…”

“…[11], put forward the conclusions of four operations rules of rough derivatives and higher order rough derivative formula but did not prove them. However, [14] only proved the four operations formulas. In the scheme of rough derivative theory Pawlak gave, this paper makes the following improvement and development for this part of context: functional features of roughly derived functions and higher order roughly derived functions are directed in rough function model.…”

“…The development and achievement of rough set theory are outstandingly manifested in various deformation and generalization of generalized rough set models [2][3][4][5][6][7][8] from the initial classical rough set model. Rough function model [9][10][11] is one of the important generalized rough set models, which are constructed by generalizing rough set theory to real numbers field and study mathematical analysis problems such as rough numbers [9,12,13] , rough continuousness [10,11,14,15] , rough calculus [10,11,[15][16][17][18][19] , rough differential equations [10,11] , etc.. Rough function model, based on rough set theory, is a more effective mathematical tool and technical means transacting indefinite and incomplete knowledge.…”

“…On rough continuity of discrete functions in rough function model, ref. [10,11,14] gave different forms of rough continuity definitions respectively. Then what is the relationship among these different forms rough continuity definitions?…”

“…Are they equivalent? Pawlak did not analyze further, while other quite few relative references [12][13][14][15][16][17][18][19] did not investigate this theory either. The completeness of related theory and applications about rough function model and the rough continuity is badly needed.…”