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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.

A $f\colon\mathbb{R}\to\mathbb{R}$ is called Hamel function if its graph is a Hamel basis of the linear space $\mathbb{R}^2$ over rationals. We construct, assuming CH, a free group of the size $2^\mathfrak{c}$ contained in the class of all Hamel functions, with the indentity function included. This answers, consistently, a question posed recently by M. Lichman, M. Pawlikowski, S. Smolarek, and J. Swaczyna., Comment: The result is false and it can't be corrected

Let X be an uncountable Polish space. Lubica Hola showed recently that there are 2^continuum many quasi-continuous real valued functions defined on the uncountable Polish space that are not Borel measurable. Inspired by Hola's result, we are extending it in two directions. First, we prove that the same conclusion holds when X is an uncountable Polish space and Y is any Hausdorff space with |Y|>1 then the family of all non-Borel measurable quasi-continuous functions has cardinality at least 2^continuum. Secondly, we show that the family of quasi-continuous non Borel functions may contain big algebraic structures., Comment: 13 pages

Let $f\colon\mathbb{R}^2\to\mathbb{R}$. The notions of feebly continuity and very feebly continuity of $f$ at a point $\langle x,y\rangle\in\mathbb{R}^2$ were considered by I. Leader in 2009. We study properties of the sets $FC(f)$ (respectively, $VFC(f)\supset FC(f)$) of points at which $f$ is feebly continuous (very feebly continuous). We prove that $VFC(f)$ is densely nonmeager, and, if $f$ has the Baire property (is measurable), then $FC(f)$ is residual (has full outer Lebesgue measure). We describe several examples of functions $f$ for which $FC(f)\neq VFC(f)$. Then we consider the notion of two-feebly continuity which is strictly weaker than very feebly continuity. We prove that the set of points where (an arbitrary) $f$ is two-feebly continuous forms a residual set of full outer measure. Finally, we study the existence of large algebraic structures inside or outside various sets of feebly continuous functions.

The characterization of the pointwise limits of the sequences of \'Swi\k{a}tkowski functions is given. Modifications of \'Swi\k{a}tkowski property with respect to different topologies finer than the Euclidean topology are discussed.