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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Mean value theorems for generalized Riemann derivatives

Authors: J. M. Ash and R. L. Jones
Journal: Proc. Amer. Math. Soc. 101 (1987), 263-271
MSC: Primary 26A24; Secondary 65D25
MathSciNet review: 902539
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Abstract: Let $ x,e \geqslant 0,{u_0} < \cdots < {u_{d + e}}$ and $ h > 0$ be real numbers. Let $ f$ be a real valued function and let $ \Delta (h;u,w)f(x){h^{ - d}}$ be a difference quotient associated with a generalized Riemann derivative. Set $ I = (x + {u_0}h,x + {u_{d + e}}h)$ and let $ f$ have its ordinary $ (d - 1)$st derivative continuous on the closure of $ I$ and its $ d$th ordinary derivative $ {f^{(d)}}$ existent on $ I$. A necessary and sufficient condition that a difference quotient satisfy a mean value theorem (i.e., that there be a $ \xi \in I$ such that the difference quotient is equal to $ {f^{(d)}}(\xi ))$ is given for $ d = 1$ and $ d = 2$. The condition is sufficient for all $ d$. It is used to show that many generalized Riemann derivatives that are "good" for numerical analysis do not satisfy this mean value theorem.

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Keywords: Generalized Riemann derivative, mean value theorem
Article copyright: © Copyright 1987 American Mathematical Society