The Hausdorff Dimension of Graphs of Prevalent Continuous Functions

Abstract

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on \([0,1]^d\) for \(d \in \mathbb{N}\) and use this to obtain results on the ‘horizon problem’ for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function.