Condition $\mathcal {B}$ and Baire $1$ generalized derivatives
Authors:
Udayan B. Darji, Michael J. Evans and Richard J. O’Malley
Journal:
Proc. Amer. Math. Soc. 123 (1995), 1727-1736
MSC:
Primary 26A24
DOI:
https://doi.org/10.1090/S0002-9939-1995-1254835-4
MathSciNet review:
1254835
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Abstract | References | Similar Articles | Additional Information
Abstract: Ordered pairs (F, f) of real-valued functions on [0,1] which satisfy the condition that every perfect set M contains a dense ${G_\delta }$ set K such that $F\backslash M$ is differentiable to f on K are shown to play a key role in several types of generalized differentiation. In particular, this condition is utilized to prove the equivalence of selective differentiation and various forms of path differentiation under the assumption that the derivatives involved are of Baire class 1, thereby providing an affirmative answer, for Baire 1 selective derivatives, to a question raised in [Trans. Amer. Math. Soc 283 (1984), 97-125].
- A. M. Bruckner, R. J. O’Malley, and B. S. Thomson, Path derivatives: a unified view of certain generalized derivatives, Trans. Amer. Math. Soc. 283 (1984), no. 1, 97–125. MR 735410, DOI https://doi.org/10.1090/S0002-9947-1984-0735410-1
- M. Laczkovich, On the Baire class of selective derivatives, Acta Math. Acad. Sci. Hungar. 29 (1977), no. 1-2, 99–105. MR 437691, DOI https://doi.org/10.1007/BF01896471
- R. J. O’Malley, Selective derivates, Acta Math. Acad. Sci. Hungar. 29 (1977), no. 1-2, 77–97. MR 437690, DOI https://doi.org/10.1007/BF01896470
- Udayan B. Darji and Michael J. Evans, Recovering Baire $1$ functions, Mathematika 42 (1995), no. 1, 43–48. MR 1346670, DOI https://doi.org/10.1112/S0025579300011335
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Keywords:
Baire class 1,
selections,
path systems,
differentiation,
Condition <!– MATH $\mathcal {B}$ –> <IMG WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathcal {B}$">
Article copyright:
© Copyright 1995
American Mathematical Society