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On sets determined by sequences of quasi-continuous functions. (English) Zbl 1002.26004
Let $X$ be a topological space. $f: X\to\bbfR$ is said to be quasi-continuous iff for $p\in X$ and open sets $U\subset X$, $W\subset\bbfR$ such that $p\in U$, $f(p)\in W$, there is an open set $G\ne\emptyset$ such that $G\subset U$, $f(G)\subset W$. The main problem of the paper is to find conditions, for given sets $L_0$, $L_{+\infty}$, $L_{-\infty}$, assuring that there exists a sequence of quasi-continuous functions $f_n$ that is convergent on $L_0$, $f_n\to+\infty$ on $L_{+\infty}$ and $f_n\to-\infty$ on $L_{-\infty}$.

26A15Continuity and related questions (one real variable)
26A21Classification of functions of one real variable; Baire classification
54C30Real-valued functions on topological spaces
54C08Weak and generalized continuity
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