Graphs of continuous functions and fractal dimension

Abstract

In this paper, we show that, for any β ∈ [1, 2], a given strictly positive real-valued continuous function on [0, 1] whose graph has upper box-counting dimension less than or equal to β can be decomposed as a product of two real-valued continuous functions on [0, 1] whose graphs have upper box-counting dimension equal to β. We also obtain a formula for the upper box-counting dimension of every element of a ring of polynomials in finite number of continuous functions on [0, 1] over the field R.