Point-set games and functions with the hereditary small oscillation property

Given a metric space X, we consider certain families of functions $f:X\to R$ 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, $\bar{\mu }$-measurable functions (for a finite nonatomic Borel measure μ 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.