具有无界变差的连续函数研究进展

Some Remarks on Continuous Functions of Unbounded Variation

讨论了具有无界变差的连续函数的结构.首先按照局部结构和分形维数对连续函数进行了分类,给出了相应的例子.对这些具有无界变差的函数的性质进行了初步的讨论.对于新定义的奇异连续函数,给出了一个等价判别定理.基于奇异连续函数,又给出了局部分形函数和分形函数的定义.同时,分形函数又由奇异分形函数、非正则分形函数和正则分形函数组成.相应于不连续函数的情形也进行了简单的讨论.

Different type of continuous functions of unbounded variation have been discussed. According to the local structure, all continuous functions have been clas- sified. Meanwhile, certain examples of different kinds of fraetal functions have been given. Properties of those fractal functions have also been discussed. Definition of singular continuous functions has been given and an equivalent theorem which can be used to describe whether a function is a singular continuous function or not has been set up. Based on definition of singular continuous functions, we make research on defi- nition of local fractal functions and fractal functions. Fractal functions are composed of singular fractal functions, irregular fractal functions and regular fractal functions. Fi- nally, definition of fractal functions corresponding to discontinuous functions has been explored.