Fractal dimensions of fractional integral of continuous functions

In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f(x) of order v(v>0) which is written as D^-vf(x) has been proved to still be continuous and bounded. Furthermore, upper box dimension of D^-vf(x) is no more than 2 and lower box dimension of D^-vf(x) is no less than 1. If f(x) is a Lipshciz function, D^-vf(x) also is a Lipshciz function. While f(x) is differentiable on0, 1], D^-vf(x) is differentiable on0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f(x) satisfying Hölder condition, upper box dimension of Riemann-Liouville fractional integral of f(x) seems no more than upper box dimension of f(x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying Hölder condition of order v(v>0) is strictly less than 2-v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.