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On sets of discrete convergence points of sequences of real functions. (English) Zbl 1070.26005
For a sequence $(a_n)$ of real numbers, $a\in\bbfR$ is said to be the discrete limit of $(a_n)$ iff there exists $k\in\bbfN$ such that $a_n= a$ for $k< n$ [cf. {\it Á. Császár} and {\it M. Laczkovic}, Studia Sci. Math. Hung. 10, 463--472 (1975; Zbl 0405.26006)]. The purpose of the paper is to characterize, for different families ${\Cal F}$ of functions $f: \bbfR\to\bbfR$, the set $L^d({\Cal F})$ of points $x\in\bbfR$, where given a sequence $(f_n)\subset{\Cal F}$, $f_n(x)$ discretely converges to some limit $f(x)$. As ${\Cal F}$, Baire class $\alpha$, Darboux functions, measurable functions, derivatives, approximately continuous functions, quasi-continuous functions, etc. are considered.
26A21Classification of functions of one real variable; Baire classification
26A03Elementary topology of the real line
54C50Special sets of topological spaces defined by functions