Typical behavior of lower scaled oscillation

Abstract

For a mapping $f\colon X\to Y$ between metric spaces the function $\text{lip} f\colon X\to[0,\infty]$ defined by $\text{lip} f(x)=\liminf_{r\to0}\frac{\text{diam} f(B(x,r))}{r}$ is termed the lower scaled oscillation or little lip function. We prove that, given any positive integer $d$ and a locally compact set $\Omega\subseteq\mathbb{R}^d$ with a nonempty interior, for a typical continuous function $f\colon \Omega\to\mathbb{R}$ the set $\{x\in\Omega:\mbox{lip} f(x)>0\}$ has both Hausdorff and lower packing dimensions exactly $d-1$, while the set $\{x\in\Omega:\text{lip} f(x)=\infty\}$ has non-$\sigma$ finite $(d{-}1)$-dimensional Hausdorff measure. This sharp result roofs previous results of Balogh and Cs\"ornyei, Hanson and Buczolich, Hanson, Rmoutil and Z\"urcher. It follows, e.g., that a graph of a typical function $f\in C(\Omega)$ is microscopic, and for a typical function $f\colon[0,1]\to[0,1]$ there are sets $A,B\subseteq[0,1]$ of lower packing and Hausdorff dimension zero, respectively, such that the graph of $f$ is contained in the set $A\times[0,1]\cup[0,1]\times B$.