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.
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