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# Cone-Monotone Functions: Differentiability and Continuity

Published:2005-10-01
Printed: Oct 2005
• Jonathan M. Borwein
• Xianfu Wang
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## Abstract

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: Cone-monotone functions, Aronszajn null set, directionally porous, sets, Gâteaux differentiability, separable space
 MSC Classifications: 26B05 - Continuity and differentiation questions 58C20 - Differentiation theory (Gateaux, Frechet, etc.) [See also 26Exx, 46G05]