Decomposition of approximate derivatives

O’Malley, Richard J.

doi:10.1090/S0002-9939-1978-0466446-9

Decomposition of approximate derivatives
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by Richard J. O’Malley PDF

Proc. Amer. Math. Soc. 69 (1978), 243-247 Request permission

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.

References

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Sur la derive approximative exacte

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