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The proximally continuous integrals

Part of: Real functions

Published online by Cambridge University Press:  09 April 2009

D. N. Sarkhel  and
A. K. De
D. N. Sarkhel
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India
A. K. De
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India
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Abstract

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This paper introduces certain generalization of the notions of approximate limit, continuity and derivative and of absolute continuity, of real functions, leading to generalized integrals of Perron and Denjoy types comprising the AP-integral of Burkill (1931) and Sonouchi and Utagawa (1949) and the AD-integral of Kubota (1963), respectiely. The generalizations are all substantiated by appropriate examples.

Keywords

AP-integralAD-integralapproximately continuousdensitysparse setproximally continuousDarboux continuityproximally absolutely continuousLusin's condition (N)PPr-integralPPl-integralPP-integralPD-integral

MSC classification

Secondary: 26A03: Foundations 26A24: Differentiation (functions of one variable) 26A39: Denjoy and Perron integrals, other special integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 1 , June 1981 , pp. 26 - 45
Copyright
Copyright © Australian Mathematical Society 1981

References

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