Covering functions by countably many functions from some families

Abstract

Let $\mathcal{A}$ be a nonempty family of functions from $\mathbb{R}$ to $\mathbb{R}$. A function $f:\mathbb{R}\to \mathbb{R}$ is said to be strongly countably $\mathcal{A}$-function if there is a sequence (f _n ) of functions from $\mathcal{A}$ such that $\mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right)$ (Gr(f) denotes the graph of f). If $\mathcal{A}$ is the family of all continuous functions, the strongly countable $\mathcal{A}$-functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011].

In this article, we prove that the families $\mathcal{A}\left( \mathbb{R} \right)$ of all strongly countably $\mathcal{A}$-functions are closed with respect to some operations in dependence of analogous properties of the families $\mathcal{A}$, and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.