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ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1254835
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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].

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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