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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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
Email: hfejzic@csusb.edu

Clifford E. Weil
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: weil@math.msu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11529-7
PII: S 0002-9939(2013)11529-7
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.