A note on local fractal dimensions of graphs of continuous functions

This paper mainly investigates local fractal dimensions of graphs of continuous functions based on the previous research. Initially, novel notions of left-side and right-side local fractal dimensions of graphs of continuous functions has been proposed, which can well characterize the local fractal characteristic of a continuous function on each side of a point. Further, the concept of local fractal dimensional spectrums of graphs of continuous functions has been established, and several examples of fractal functions with different local fractal dimensional spectrums have been provided to show their diversity. Besides, the application of local fractal dimensions to the fractal function approximation theory associated with unbounded variation points has been discussed. It has also been shown that there exists a sequence of continuous functions approximating the objective function which can preserve both the location of unbounded variation points and their corresponding left-side and right-side local fractal dimensions.