Decomposition of approximate derivatives

O’Malley, Richard J.

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

Decomposition of approximate derivatives
HTML articles powered by AMS MathViewer

by Richard J. O’Malley

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

DOI: https://doi.org/10.1090/S0002-9939-1978-0466446-9

PDF | 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

Casper Goffman and C. J. Neugebauer, On approximate derivatives, Proc. Amer. Math. Soc. 11 (1960), 962–966. MR 118792, DOI 10.1090/S0002-9939-1960-0118792-2
Casper Goffman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497–505. MR 137805
G. Petruska and M. Laczkovich, Baire $1$ functions, approximately continuous functions and derivatives, Acta Math. Acad. Sci. Hungar. 25 (1974), 189–212. MR 379766, DOI 10.1007/BF01901760
Richard J. O’Malley, A density property and applications, Trans. Amer. Math. Soc. 199 (1974), 75–87. MR 360955, DOI 10.1090/S0002-9947-1974-0360955-X

Sur la derive approximative exacte

4

Hassler Whitney, On totally differentiable and smooth functions, Pacific J. Math. 1 (1951), 143–159. MR 43878, DOI 10.2140/pjm.1951.1.143