Graphs of Continuous Functions from ℝ to ℝ Are Not Purely Unrectifiable

Abstract

We present an elementary proof that the graph of a continuous function from $\mathbb{R}$ to $\mathbb{R}$ is not purely unrectifiable. As a consequence of our method, we observe that all continuous functions from $\mathbb{R}$ to $\mathbb{R}$ meet the graph of some monotonic function in a set of positive linear measure.