The classification of one-dimensional continuous functions and their fractional integral

This paper mainly discusses the continuous functions whose fractal dimension is 1 on [0,1]. First, we classify continuous functions into unbounded variation and bounded variation. Then we prove that the fractal dimension of both continuous functions of bounded variation and their fractional integral is I. As for continuous functions of unbounded variation, we solve several special types. Finally, we give the example of one-dimensional continuous function of unbounded variation.