Covering functions by countably many functions from some families

Abstract

Let A be a nonempty family of functions from R to R. A function f : R → R is said to be strongly countably A-function if there is a sequence (fn) of functions from A such that Gr(f) ⊂ n Gr(fn) (Gr(f) denotes the graph of f). If A is the family of all continuous functions, the strongly countable 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 A(R) of all strongly countably A-functions are closed with respect to some operations in dependence of analogous properties of the families A, and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.