COMPOSITION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

Abstract

In this paper, we make research on composition of continuous functions with Box dimension one of bounded variation or unbounded variation on [Formula: see text]. It has been proved that one-dimensional continuous functions must be one of functions with bounded variation, or functions with finite unbounded variation points, or functions with infinite unbounded variation points on [Formula: see text]. Based on discussion of one-dimensional continuous functions, fractal dimension, such as Box dimension, of Riemann–Liouville (R-L) fractional integral of those functions have been calculated. We get an important conclusion that Box dimension of R-L fractional integral of any one-dimensional continuous functions of any positive orders still is one. R-L fractional derivative of certain one-dimensional continuous functions has been explored elementary.