Typical behavior of lower scaled oscillation

Abstract

For a mapping $f:X\to Y$ between metric spaces the function $\text{lip}f:X\to [0,\infty ]$ defined by $\text{lip}f(x)={\mathrm{liminf}}_{r\to 0}\frac{\text{diam}f(B(x,r))}{r}$ 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 $f:\Omega \to \mathbb{R}$ the set $\{x\inΩ:\mbox{lip} f(x)>0\}$ 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 Csörnyei, Hanson and Buczolich, Hanson, Rmoutil and Zürcher. It follows, e.g., that a graph of a typical function $f\in C(\Omega )$ is microscopic, and for a typical function $f:[0,1]\to [0,1]$ 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 $A\times [0,1]\cup [0,1]\times B$.