Abstract
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.
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