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On functions with summable approximate Peano derivative


Author: Cheng Ming Lee
Journal: Proc. Amer. Math. Soc. 57 (1976), 53-57
DOI: https://doi.org/10.1090/S0002-9939-1976-0399379-5
MathSciNet review: 0399379
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Abstract | References | Additional Information

Abstract: Let $ n$ be a positive integer and $ F$ a function defined on a closed interval $ I$. For $ x$ in $ I$, let the $ n$th approximate Peano derivative of $ F$ at $ x$, if it exists, be denoted as $ {F_{(n)}}(x)$. For $ n = 1$, the existence of $ {F_{(n - 1)}}(x)$ will simply mean that the function $ {F_{(0)}}( \equiv F)$ is approximately continuous at $ x$. Then the following theorem is proved, noting that the phrase ``for nearly all $ x$ in $ I$'' means ``for all $ x$ in $ I$ except perhaps for those points in a countable subset of $ I$", Theorem $ {{\mathbf{A}}_n}$. Let $ {F_{(n - 1)}}(x)$ exist finitely for all $ x$ in $ I$. If $ {F_{(n)}}(x)$ exists finitely for nearly all $ x$ in $ I$ and is summable on $ I$, then $ {F_{(n - 1)}}$ is absolutely continuous in $ I$.


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

  • [1] C. Goffman, On functions with summable derivative, Amer. Math. Monthly 78 (1971), 874-875. MR 44 #4156. MR 0286949 (44:4156)
  • [2] E. W. Hobson, The theory of functions of a real variable, 3rd ed., Cambridge, 1927, p. 359.
  • [3] C. M. Lee, On the approximate Peano derivatives, J. London Math. Soc. (to appear). MR 0399378 (53:3222)
  • [4] C. M. Lee and D. W. Solomon, On monotone functions and Perron and Denjoy integrals, Department of Mathematics, University of Wisconsin-Milwaukee (unpublished).
  • [5] J. Ridder, Ueber die genseitigen Beziehungen vershieder approximativ stetiger Denjoy-Perron-Integrale, Fund. Math. 22 (1934), 136-162.
  • [6] S. Saks, Theory of the integral, 2nd rev. ed., Monografie Mat., vol. VII, PWN, Warsaw, 1937.
  • [7] W. Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Ser. in Higher Math., McGraw-Hill, New York, 1974, pp. 179-180. MR 49 #8783. MR 0344043 (49:8783)
  • [8] T. Bagby and W. P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129-148. MR 49 #9129. MR 0344390 (49:9129)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0399379-5
Keywords: Approximate continuity, approximate Peano derivative, absolute continuity, generalized absolute continuity, monotone increasing, summable functions
Article copyright: © Copyright 1976 American Mathematical Society

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