A density property and applications
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- by Richard J. O’Malley
- Trans. Amer. Math. Soc. 199 (1974), 75-87
- DOI: https://doi.org/10.1090/S0002-9947-1974-0360955-X
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Abstract:
An unexpected metric density property of a certain type of ${F_\sigma }$ set is proven. An immediate application is a characterization of monotone functions similar to a well-known result by Zygmund. Several corollaries of this characterization are given as well as a simple proof of a theorem due to Tolstoff.References
- S. Saks, Theory of the integral, 2nd rev. ed., Monografie Mat., vol. VII, PWN, Warsaw, 1937.
- G. Tolstoff, Sur quelques propriétés des fonctions approximativement continues, Rec. Math. (Moscou) [Mat. Sbornik] N.S. 5(47) (1939), 637–645 (French, with Russian summary). MR 0001267
- Casper Goffman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497–505. MR 137805 A. Denjoy, Mémoire sur les nombres dérivés des fonctions continues, J. Math. Pures Appl. 7 (1915), 105-240.
- Casper Goffman and Daniel Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116–121. MR 120327, DOI 10.1090/S0002-9939-1961-0120327-6
- Donald Ornstein, A characterization of monotone functions, Illinois J. Math. 15 (1971), 73–76. MR 274679
- A. M. Bruckner, An affirmative answer to a problem of Zahorski, and some consequences, Michigan Math. J. 13 (1966), 15–26. MR 188375, DOI 10.1307/mmj/1028999475
- H. T. Croft, A note on a Darboux continuous function, J. London Math. Soc. 38 (1963), 9–10. MR 147588, DOI 10.1112/jlms/s1-38.1.9
- István S. Gál, On the fundamental theorems of the calculus, Trans. Amer. Math. Soc. 86 (1957), 309–320. MR 93562, DOI 10.1090/S0002-9947-1957-0093562-7 E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, Oxford, 1939.
- John L. Leonard, Some conditions implying the monotonicity of a real function, Rev. Roumaine Math. Pures Appl. 17 (1972), 757–780. MR 304574
- J. C. Burkill, The approximately continuous-Perron integral, Math. Z. 34 (1932), no. 1, 270–278. MR 1545252, DOI 10.1007/BF01180588 G. Goldowski, Note sur les dérivés exactes, Mat. Sb. 35 (1928), 35-36. G. Tolstoff, Sur la dérivé approximative exacte, Mat. Sb. 4 (1938), 499-504.
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 199 (1974), 75-87
- MSC: Primary 26A48
- DOI: https://doi.org/10.1090/S0002-9947-1974-0360955-X
- MathSciNet review: 0360955