Path derivatives and growth control
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- by A. M. Bruckner and K. G. Johnson PDF
- Proc. Amer. Math. Soc. 91 (1984), 46-48 Request permission
Abstract:
We show that a standard theorem relating the growth of a function on a measurable set to constraints on its Dini derivates extends to a class of generalized derivatives. More precisely, we show that if $\left | \bar {F}’_E \right | \leqslant M$ and $\left | \underline {F}’_E \right | \leqslant M$ on a measurable set $A$, then $\lambda (F(A)) \leqslant M\lambda (A)$. Here $\lambda$ denotes Lebesgue measure and ${\bar F’_E}$ and $\underline {F}’_E$ are the extreme derivates of $F$ relative to a system of paths which satisfy the Intersection Condition [1]. In particular, the result holds in the setting of unilateral differentiation, approximate differentiation, preponderant differentiation and qualitative differentiation.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 46-48
- MSC: Primary 26A24; Secondary 28A12
- DOI: https://doi.org/10.1090/S0002-9939-1984-0735561-7
- MathSciNet review: 735561