The quasicontinuity of delta-fine functions. (English) Zbl 1050.26002

Evans, Michael J.; Humke, Paul D.

Real Anal. Exch. 28(2002-2003), No. 2, 543-547 (2003).

Let \(f:[0,1] \to {\mathcal R}\) and let \(C(f)\) be the set of all continuity points of \(f\). If \(\tau = \{ a = a_0 < a_1 < \cdots < a_n = b\} \) is a partition of \([a,b]\) then \(P_{f,\tau ,[a,b]}\) denotes that function which agrees with \(f\) at points of \(\tau \) and is linear on the intervening intervals. A function \(f\) is said to have the delta-fine property if for each closed set \(W\) and each \(\eta > 0\), there are two points \(a < b\) in \(C(f)\) such that \([a,b]\cap W \neq \emptyset \) and such that for every \(\delta > 0\) there is a \(\delta \)-fine partition \(\tau \) of \([a,b]\) consisting of points in \(C(f)\) with \(| P_{f,\tau ,[a,b]}(x) - f(x)| < \eta \) for \(x \in W\cap [a,b]\). In this article the authors prove that if \(f\) has the delta-fine property then for each \(\eta > 0\) the set \(NQ(f)\) of all points where \(f\) is not quasicontinuous is the union of a countable family of \((1- \eta )\)-symmetrically porous sets.

Reviewer: Zbigniew Grande (Bydgoszcz)

MSC:

26A21	Classification of real functions; Baire classification of sets and functions
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

universally polygonally approximable function; delta-fine; quasicontinuity