On the fractal dimensions of continuous functions and algebraic genericity

Abstract

In this paper, we consider the algebraic genericity of continuous functions in terms of Hausdorff and box dimensions. More precisely, we show that, given $s\in (1,2]$, the set of functions $f\in C[0,1]$ whose graphs have box and Hausdorff dimensions s everywhere in the unit interval is maximal-dense-lineable and also dense-algebrable. This work answers affirmatively two questions of Bonilla et al. (2019) [15]. In addition, we consider the algebraic genericity of continuous functions whose level sets satisfy some dimensional properties.