This paper considers the relationship of box dimension between a continuous fractal function and its Riemann–Liouville fractional integral. For an arbitrary fractal function [Formula: see text] it is proved that the upper box dimension of the graph of Riemann–Liouville fractional integral [Formula: see text] does not exceed the upper box dimension of [Formula: see text], i.e. [Formula: see text]. This estimate shows that [Formula: see text]order Riemann–Liouville fractional integral [Formula: see text] does not increase the fractal dimension of the integrand [Formula: see text], which means that Riemann–Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229] and [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438].
In the present paper, a one-dimensional continuous function of unbounded variation on the interval [0, 1] has been constructed. Box dimension of this function has been proved to be 1. Furthermore, Box dimension of its Riemann-Liouville fractional integral of any order has also been proved to be 1.
Let [Formula: see text] be [Formula: see text]-Hölder continuous on [Formula: see text] and well-defined about the Weyl fractional integral. Then, [Formula: see text] where [Formula: see text] and [Formula: see text]. This estimation shows that the Box dimension of [Formula: see text] leads to some similar linear dimension decrease like the Riemann–Liouville fractional integral [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Sin. 32 (2016) 1494–1508].
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