A typical property of Baire Darboux functions
Authors:
Michael J. Evans and Paul D. Humke
Journal:
Proc. Amer. Math. Soc. 98 (1986), 441447
MSC:
Primary 26A27; Secondary 26A21
MathSciNet review:
857937
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Abstract: It is well known that a realvalued, 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 nonporous 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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 A. M. Bruckner, Differentiation of real functions, Lecture Notes in Math., vol. 659, SpringerVerlag, Berlin and New York, 1978. MR 507448 (80h:26002)
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 J. Ceder and T. Pearson, A survey of Darboux Baire 1 functions, Real Anal. Exchange 9 (198384), 179194. MR 742783 (86a:26003)
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 E. P. Dolzenko, Boundary properties of arbitrary functions, Math. USSRIzv. 1 (1967), 112. MR 0217297 (36:388)
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 M. J. Evans and P. D. Humke, Approximate continuity points and points of integrable functions, Real Anal. Exchange 11 (198586), 390410. MR 844259 (87j:26012)
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 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. 8186. MR 807980 (86m:26004)
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 J. Tkadlec, Construction of some nonporous sets on the real line, Real Anal. Exchange 9 (198384), 473482. MR 766073 (85m:26001)
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 W. H. Young, A theorem in the theory of functions of a real variable, Rend. Circ. Mat. Palermo 24 (1907), 187192.
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 L. Zajicek, Sets of porosity and sets of porosity , Časopis Pěst. Mat. 101 (1976), 350359. MR 0457731 (56:15935)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198608579371
PII:
S 00029939(1986)08579371
Article copyright:
© Copyright 1986
American Mathematical Society
