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 be a positive integer and a function defined on a closed interval . For in , let the th approximate Peano derivative of at , if it exists, be denoted as . For , the existence of will simply mean that the function is approximately continuous at . Then the following theorem is proved, noting that the phrase ``for nearly all in '' means ``for all in except perhaps for those points in a countable subset of ", Theorem . Let *exist finitely for all in . If exists finitely for nearly all in and is summable on , then is absolutely continuous in* .

**[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monotone functions and Perron and Denjoy integrals, Department of Mathematics*, University of Wisconsin-Milwaukee (unpublished).**[5]**J. Ridder,*Ueber die genseitigen Beziehungen vershieder approximativ stetiger Denjoy-Perron-Integrale*, Fund.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