FRACTAL DIMENSIONS OF LINEAR COMBINATION OF CONTINUOUS FUNCTIONS WITH THE SAME BOX DIMENSION

In this paper, we mainly discuss continuous functions with certain fractal dimensions on [Formula: see text]. We find space of continuous functions with certain Box dimension is not closed. Furthermore, Box dimension of linear combination of two continuous functions with the same Box dimension maybe does not exist. Definitions of fractal functions and local fractal functions have been given. Linear combination of a fractal function and a local fractal function with the same Box dimension must still be the original Box dimension with nontrivial coefficients.