Variable-Order Riemann-Liouville Integrals on the Box Dimension of Graphs of Continuous Functions

This paper investigates the effect of the variable-order Riemann-Liouville fractional integral operator on the box dimension of graphs of continuous functions. By constructing a counterexample, we show that even when the integrand is a constant function, the graph of the resulting integral has box dimension strictly greater than 1 if the order function exhibits fractal characteristics, revealing an essential difference between variable-order and constant-order fractional operators. On the other hand, we provide a sufficient condition ensuring that the output graph has box dimension equal to 1: if the order function is Lipschitz continuous with a positive lower bound and the integrand satisfies an appropriate H¨older continuity condition, then the resulting integral belongs to the class of absolutely continuous functions, and hence its graph has box dimension 1. Numerical simulations illustrate the conclusion of the counterexample.