The Hausdorff dimension of graphs of prevalent continuous functions

TL;DR

This paper establishes that the Hausdorff dimension of the graph of a prevalent continuous function is 2 and extends these results to higher dimensions, contributing to the understanding of fractal surfaces and the horizon problem.

Contribution

It proves the Hausdorff dimension is 2 for prevalent continuous functions and extends the results to functions on higher-dimensional domains.

Findings

Hausdorff dimension of prevalent functions' graphs is 2

Extension of results to functions on [0,1]^d

Applications to the horizon problem for fractal surfaces

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.