Typical Approximately Continuous Functions Are Surprisingly Thick

Abstract

We show that the following properties are typical ones in the space of bounded approximately continuous functions: The Hausdorff dimension of the graph of $f$ equals two and any level set $f^{-1}\left(y\right)$ for inf sup $f$ has Hausdorff dimension one. This example serves as a warning to be careful when applying plausible but unproved “interpolation principles”.