On the nondifferentiability of cone-monotone functions in Banach spaces

  • Jonathan BorweinEmail author
  • Rafal Goebel
Part of the Springer Optimization and Its Applications book series (SOIA, volume 32)


In finite-dimensional spaces, cone-monotone functions – a special case of which are coordinate-wise nondecreasing functions – possess several regularity properties like almost everywhere continuity and differentiability. Such facts carry over to a separable Banach space, provided that the cone has interior. This chapter shows that further generalizations are not readily possible. We display several examples of cone–monotone functions on various Banach spaces, lacking the regularity expected from their finite-dimensional counterparts.


Monotone functions ordered Banach spaces generating cones differentiability 


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The first author’s research was partially supported by NSERC and by the Canada Research Chair Programme. The second author performed this research at the Centre for Experimental and Constructive Mathematics at Simon Fraser University and at the Department of Mathematics at University of British Columbia.


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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Centre for Experimental and Constructive MathematicsSimon Fraser UniversityBurnabyCanada V5A 1S6
  2. 2.Center for Control Engineering and Computation ECEUniversity of CaliforniaSanta BarbaraU. S. A

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