Graphs of continuous functions and fractal dimension

Abstract

In this paper, we show that, for any $\beta \in [1,2]$, a given strictly positive real-valued continuous function on $[0,1]$ whose graph has upper box-counting dimension less than or equal to $\beta $ can be decomposed as a product of two real-valued continuous functions on $[0,1]$ whose graphs have upper box-counting dimension equal to $\beta$. 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 $\mathbb{R}.$