Comparable almost continuous functions. (Q595848)

The author defines three conditions for families of real valued functions. The first one characterizes those families \(\mathcal F\) of size \(\leq\) continuum for which there is a positive function \(g\) such that \(f+g\) is almost continuous in the sense of Stallings. The second one describes the same property for families \(\mathcal F\) of size less than the cofinality of continuum. The last one characterizes the family of functions which are the averages of two comparable almost continuous functions. The same characterizations are valid also if one replaces almost continuous functions, by Darboux functions (see also [\textit{A. Maliszewski}, Trans. Am. Math. Soc. 350, No.~7, 2833--2846 (1998; Zbl 0908.26005)]).