THE FRACTAL DIMENSION GRAPHS OF CERTAIN FRACTAL FUNCTIONS

In this paper, we preliminarily investigate continuous functions of graphs with fractal characteristics. First, definitions of the local fractal function and the global fractal function are provided. They are classified according to their local structure, namely, regular fractal functions, nonregular fractal functions, and singular fractal functions. Second, definitions of bounded and unbounded variation points have been given. Based on definitions of fractal dimensions and variation of points, we provide definition of the single point fractal dimension. Third, the fractal dimension obtained point by point can form the fractal dimension graph of the corresponding fractal function. The fractal dimension graph can better help characterize fractal characteristics of fractal functions, especially those with the same fractal dimension. At the same time, we provide specific examples of fractal functions and their fractal dimension graphs to illustrate.