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On quasi-uniform convergence of a sequence of s. q. c. functions. (English) Zbl 0961.26004
In [Z. Grande, Math. Slovaca 44, No. 3, 297-301 (1994; Zbl 0837.26003)] it is proved that every real cliquish (i.e., the set $$D(f)$$ of all discontinuity points of $$f$$ is of the first category) function $$f: \mathbb R\to \mathbb R$$ is the quasiuniform limit of a sequence of Darboux quasi-continuous functions. In this paper it is shown an analogous result for the measure case: every almost continuous (i.e., $$D(f)$$ is of measure zero) function $$f: \mathbb R\to \mathbb R$$ is the quasi-uniform limit of a sequence of Darboux strongly quasi-continuous functions. (A function $$f: \mathbb R\to \mathbb R$$ is said to be strongly quasi-continuous (briefly s.q.c.) at a point $$x$$ if for every set $$A$$ containing $$x$$, which is open in the density topology, and for every positive real $$\eta$$, there is an open interval $$I$$ such that $$I\cap A\neq \emptyset$$ and $$|f(t)-f(x)|<\eta$$ for all $$t\in I\cap A$$).
##### MSC:
 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 54C30 Real-valued functions in general topology 54C08 Weak and generalized continuity
Zbl 0837.26003
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##### References:
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