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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The pointwise characterization of derivatives of integrals

Author: D. N. Sarkhel
Journal: Proc. Amer. Math. Soc. 63 (1977), 125-128
MSC: Primary 28A15
MathSciNet review: 0437703
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Abstract: Let $ \sigma $ denote the indefinite integral of a summable function f on $ {R_n}$. We give necessary and sufficient conditions for $ \sigma $ to possess a strong, or ordinary, or general derivative, equal to $ f(x)$ at a point x of approximate continuity of f. Munroe [3] states that when f is a bounded function then $ f(x)$ is the strong derivative of $ \sigma $ at x if and only if x is a point of approximate continuity of f. We point out an error in the proof of the only if part of this result and show by example that this part of Munroe's result is in fact false.

References [Enhancements On Off] (What's this?)

  • [1] H. Busemann and W. Feller, Zur Differentiation der Lebesgueschen Integrale, Fund. Math. 22 (1934), 226-256.
  • [2] B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935),217-234.
  • [3] M. E. Munroe, Introduction to measure and integration, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1953. MR 0053186
  • [4] Stanisław Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. MR 0167578

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Keywords: Euclidean n-space, strong derivative, general derivative, ordinary derivative, density, approximate continuity, summable
Article copyright: © Copyright 1977 American Mathematical Society