Path derivatives and growth control
Authors: A. M. Bruckner and K. G. Johnson
Journal: Proc. Amer. Math. Soc. 91 (1984), 46-48
MSC: Primary 26A24; Secondary 28A12
MathSciNet review: 735561
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 and on a measurable set , then . Here denotes Lebesgue measure and and are the extreme derivates of relative to a system of paths which satisfy the Intersection Condition . In particular, the result holds in the setting of unilateral differentiation, approximate differentiation, preponderant differentiation and qualitative differentiation.
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Keywords: Generalized derivatives, path derivatives, Intersection Condition, generalized absolute continuity
Article copyright: © Copyright 1984 American Mathematical Society