Covering functions by countably many functions from some families (Q889466)

The concept of countable continuity is most known in the setting of the \textit{Baire one star property} [\textit{R. J. O'Malley}, Proc. Am. Math. Soc. 60, 187--192 (1977; Zbl 0339.26010)] in which the assumption on \(f:\mathbb R\to\mathbb R\) says that the domain can be partitioned into countably many closed sets \(B_n\) so that every restriction \(f{\restriction}B_n\) is a continuous function. A much weaker condition one obtains without the assumption that \(B_n\) is closed. A middle variant has been proposed by the author some time ago: \(f\) is said to be \textit{strongly countably continuous} if the graph of \(f\) can be covered by countably many graphs of continuous functions [\textit{Z. Grande} and \textit{A. Fatz-Grupka}, Tatra Mt. Math. Publ. 28, No. 1, 57--63 (2004; Zbl 1111.26005)]. Unlike for Baire one star property, Grande's strong countable continuity does not agree much with Baire or Borel classification since, obviously, strongly countably continuous functions can be neither Lebesgue nor Baire measurable. On the other hand, not all approximately continuous functions (so, not all Baire one functions) are strongly countably continuous [\textit{G. Horbaczewska}, Tatra Mt. Math. Publ. 42, 81--86 (2009; Zbl 1212.26003)]. In the present work, the author considers, instead of continuous functions, other classes of functions whose graphs are to cover the graph of \(f\): differentiable functions (section 2.2), approximately continuous functions (2.3), and quasi-continuous functions (2.4), thus giving rise to the notions of respectively strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions. The main results consist of showing that every Baire one function is strongly countably approximately continuous and every function with the Baire property is strongly countably quasi-continuous.