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A property of Peano derivatives in several variables

Authors: Hajrudin Fejzić and Clifford E. Weil
Journal: Proc. Amer. Math. Soc. 141 (2013), 2411-2417
MSC (2010): Primary 26B05, 26B35
Published electronically: March 22, 2013
MathSciNet review: 3043022
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Abstract: Let $ f$ be a function of several variables that is $ n$ times Peano differentiable. Andreas Fischer proved that if there is a number $ M$ such that $ f_{\boldsymbol {\alpha } } \ge M$ or $ f_{\boldsymbol {\alpha } } \le M$ for each $ \boldsymbol {\alpha }$, with $ \left \vert \boldsymbol {\alpha } \right \vert = n $, then $ f$ is $ n$ times differentiable in the usual sense. Here that result is improved to permit the type of one-sided boundedness to depend on $ \boldsymbol {\alpha }$.

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Additional Information

Hajrudin Fejzić
Affiliation: Department of Mathematics, California State University, San Bernardino, California 92407-2397

Clifford E. Weil
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Keywords: Peano derivatives, several variables
Received by editor(s): August 13, 2011
Received by editor(s) in revised form: October 20, 2011
Published electronically: March 22, 2013
Additional Notes: The first author was supported in part by CSUSB 2011 Summer Research Grant
Communicated by: Tatiana Toro
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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