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On sets of discrete convergence points of sequences of real functions. (English) Zbl 1070.26005
For a sequence $\left({a}_{n}\right)$ of real numbers, $a\in ℝ$ is said to be the discrete limit of $\left({a}_{n}\right)$ iff there exists $k\in ℕ$ such that ${a}_{n}=a$ for $k [cf. Á. Császár and M. Laczkovic, Studia Sci. Math. Hung. 10, 463–472 (1975; Zbl 0405.26006)]. The purpose of the paper is to characterize, for different families $ℱ$ of functions $f:ℝ\to ℝ$, the set ${L}^{d}\left(ℱ\right)$ of points $x\in ℝ$, where given a sequence $\left({f}_{n}\right)\subset ℱ$, ${f}_{n}\left(x\right)$ discretely converges to some limit $f\left(x\right)$. As $ℱ$, Baire class $\alpha$, Darboux functions, measurable functions, derivatives, approximately continuous functions, quasi-continuous functions, etc. are considered.
##### MSC:
 26A21 Classification of functions of one real variable; Baire classification 26A03 Elementary topology of the real line 54C50 Special sets of topological spaces defined by functions