Characterization of Fractal Features of the union of a class of function graphs

The union of a class of function graphs and their fractal characteristics has been discussed. Research has shown that smooth curves can be listed and form graphs of fractal functions. For example, for the power function sequence [Formula: see text] on [0, 1/2], the Box dimension of the union of their graphs is 1. For the alternative sequence [Formula: see text] on [0, 1/2], the Box dimension of the corresponding graph union is 1/2. The set of intersection points between the corresponding graph and a line parallel to the coordinate axis will also generate a fractal point set of non-integer dimensions. Our results reveal that the convergence rate of the function sequence at the boundary directly affects the fractal dimension of its graph union, providing a new perspective on the relationship between fractal geometry and function sequences.