Some Integrals Involving the Cantor Function

Many mathematicians are familiar with the Cantor set and the Cantor function since they both provide good counterexamples for simplistic thinking about sets of real numbers and real-valued functions. The Cantor function is a continuous, nondecreasing, nonconstant function defined on (0, 1) whose der...

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Veröffentlicht in:The American mathematical monthly 2009-03, Vol.116 (3), p.218-227
1. Verfasser: Gordon, Russell A.
Format: Artikel
Sprache:eng
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Zusammenfassung:Many mathematicians are familiar with the Cantor set and the Cantor function since they both provide good counterexamples for simplistic thinking about sets of real numbers and real-valued functions. The Cantor function is a continuous, nondecreasing, nonconstant function defined on (0, 1) whose derivative is zero at every point in the complement of the Cantor set, that is, almost everywhere. It is thus a continuous function of bounded variation that is not absolutely continuous. Here, Gordon shows that the value of an expression for ∫subosup1 f [order of] c, where f is a continuous function, is closely related to the midpoint rule approximations of ∫ssub0sup1 f.
ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2009.11920931