On a. c. limits and monotone limits of sequences of jump functions (Q1852337)

If \(A\) is a class of functions \(f:\mathbb{R}\to\mathbb{R}\), let \(B^*_1(A)\) denote the class of those \(f:\mathbb{R}\to\mathbb{R}\) for which there is a sequence \(f_n\in A\) such that, given \(x\in\mathbb{R}\), there is \(k\in\mathbb{N}\) for which \(f_n(x)= f(x)\) whenever \(n> k\) [cf. \textit{Á. Császár} and \textit{M. Laczkovich}, Stud. Sci. Math. Hung. 10, 463-472 (1975; Zbl 0405.26006)]. Let \(P\) denote the class of all \(f:\mathbb{R}\to\mathbb{R}\) for which the unilateral limits \(\lim_{t\to x+}f(t)\) and \(\lim_{t\to x-}f(t)\) both exist and are finite for every \(x\in\mathbb{R}\). In the paper, necessary and sufficient conditions for \(f\in B^*_1(P)\) are given. The case of monotone sequences \(f_n\in P\) is also considered.