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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Decomposition of approximate derivatives


Author: Richard J. O’Malley
Journal: Proc. Amer. Math. Soc. 69 (1978), 243-247
MSC: Primary 26A24
MathSciNet review: 0466446
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Abstract: It is shown that if $ f:[0,1] \to R$ has a finite approximate derivative $ {f'_{{\text{ap}}}}$ everywhere in [0, 1], then there is a sequence of perfect sets $ {H_n}$, whose union is [0, 1], and a sequence of differentiable functions, $ {h_n}$, such that $ {h_n} = f$ over $ {H_n}$ and $ {h'_n} = {f'_{{\text{ap}}}}$ over $ {H_n}$. This result follows from a new, more general theorem relating approximate differentiability and differentiability. Applications of both theorems are given.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0466446-9
PII: S 0002-9939(1978)0466446-9
Keywords: Points of density, approximate derivative, Baire category theorem
Article copyright: © Copyright 1978 American Mathematical Society