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Cone-Monotone Functions: Differentiability and Continuity

Published online by Cambridge University Press:  20 November 2018

Jonathan M. Borwein and
Xianfu Wang
Jonathan M. Borwein
Affiliation:
Faculty of Computer Science, Dalhousie University, 6050 University Avenue, Halifax, NS, B3H 1W5, e-mail: jborwein@cs.dal.ca
Xianfu Wang
Affiliation:
Department of Mathematics and Statistics, Okanagan University College, Kelowna, BC, V1V 1V7, e-mail: xwang@ouc.bc.ca
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Abstract

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We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$ -monotone functions has a $\sigma $ -porous complement in the space of continuous functions endowed with the uniform metric.

Keywords

Primary: 26B05secondary: 58C20Cone-monotone functionsAronszajn null setdirectionally porous setsGâteaux differentiabilityseparable spaces

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 5 , 01 October 2005, pp. 961 - 982
Copyright
Copyright © Canadian Mathematical Society 2005

References

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