Families of feebly continuous functions and their properties

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.