  Given a metric space $X$, we consider certain families of functions
$f:X\to\mathbb{R}$ having the hereditary oscillation property HSOP and the
hereditary continuous restriction property HCRP on large sets. When $X$ is
Polish, among them there are families of Baire measurable functions,
$\overline{\mu}$-measurable functions (for a finite nonatomic Borel measure
$\mu$ on $X$) and Marczewski measurable functions. We obtain their
characterizations using a class of equivalent point-set games. In similar
aspects, we study cliquish functions, SZ-functions and countably continuous
functions.
