In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions.

在本文中，我们主要研究在闭区间上具有无限变化的连续函数。鉴于在分形函数范围内对具有分形结构的不同类型函数的提议和定义越来越多，引入系统的分类并讨论分形函数的基本定义是很重要的。已经给出了具有某些分形维数的分形函数的不同示例。然后，我们介绍了不同类型的分形函数，如奇异分形函数、不规则分形函数、正则分形函数和完全正则分形函数。我们还分别介绍了具有一维和二维的整数维分形函数。已经给出了具有分形结构的其他连续和不连续函数。研究了分形函数在插值、逼近和分数微积分之间的关​​系中的应用。最后，我们研究了具有不同分形维数的分形维空间。