On singularity of Henstock integrable functions. (English) Zbl 1015.26016

Lu, Jitan; Lee, Peng-Yee

Real Anal. Exch. 25(1999-2000), No. 2, 795-797 (2000).

If \(f : [0,1]\to\mathbb R\) is Henstock–Kurzweil integrable then \(x\in[0,1]\) is a point of non-summability (‘singular point’ in the paper under review) if \(\int_I|f|\) diverges for every open interval \(x\in I\subset[0,1]\). An example shows that for each \(0<\lambda<1\) there is a Henstock–Kurzweil integrable function \(f\) such that the set of points of non-summability has measure \(\lambda\).
All the results of this paper, including the definition of point of non-summability and the example, are contained in pages 147-149 of [R. L. Jeffery, “The theory of functions of a real variable” (1951; Zbl 0043.27901)].

Reviewer: Erik O.Talvila (Edmonton, AB)

MSC:

26A39

Denjoy and Perron integrals, other special integrals

Keywords:

singular point; Henstock-Kurzweil integrable function; point of non-summability

Citations:

Zbl 0043.27901