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A typical property of Baire $ 1$ Darboux functions


Authors: Michael J. Evans and Paul D. Humke
Journal: Proc. Amer. Math. Soc. 98 (1986), 441-447
MSC: Primary 26A27; Secondary 26A21
DOI: https://doi.org/10.1090/S0002-9939-1986-0857937-1
MathSciNet review: 857937
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Abstract: It is well known that a real-valued, bounded, Baire class one function of a real variable is the derivative of its indefinite integral at every point except possibly those in a set which is both of measure zero and of first category. In the present paper, a bounded, Darboux, Baire class one function is constructed to have the property that its indefinite integral fails to be differentiable at non-$ \sigma $-porous set of points. Such functions are then shown to be "typical" in the sense of category in several standard function spaces.


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  • [1] A. M. Bruckner, Differentiation of real functions, Lecture Notes in Math., vol. 659, Springer-Verlag, Berlin and New York, 1978. MR 507448 (80h:26002)
  • [2] J. Ceder and T. Pearson, A survey of Darboux Baire 1 functions, Real Anal. Exchange 9 (1983-84), 179-194. MR 742783 (86a:26003)
  • [3] E. P. Dolzenko, Boundary properties of arbitrary functions, Math. USSR-Izv. 1 (1967), 1-12. MR 0217297 (36:388)
  • [4] M. J. Evans and P. D. Humke, Approximate continuity points and $ L$-points of integrable functions, Real Anal. Exchange 11 (1985-86), 390-410. MR 844259 (87j:26012)
  • [5] P. D. Humke and B. S. Thomson, A porosity characterization of symmetric perfect sets, Classical Real Analysis, Contemporary Math., vol. 42, Amer. Math. Soc., Providence, R.I., 1985, pp. 81-86. MR 807980 (86m:26004)
  • [6] J. Tkadlec, Construction of some non-$ \sigma $-porous sets on the real line, Real Anal. Exchange 9 (1983-84), 473-482. MR 766073 (85m:26001)
  • [7] W. H. Young, A theorem in the theory of functions of a real variable, Rend. Circ. Mat. Palermo 24 (1907), 187-192.
  • [8] L. Zajicek, Sets of $ \sigma $-porosity and sets of $ \sigma $-porosity $ (q)$, Časopis Pěst. Mat. 101 (1976), 350-359. MR 0457731 (56:15935)

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DOI: https://doi.org/10.1090/S0002-9939-1986-0857937-1
Article copyright: © Copyright 1986 American Mathematical Society

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