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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
Citations:
Zbl 0837.26003
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References:
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