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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 note on the differentiability of convex functions


Authors: Cong Xin Wu and Li Xin Cheng
Journal: Proc. Amer. Math. Soc. 121 (1994), 1057-1062
MSC: Primary 46G05; Secondary 49J50
MathSciNet review: 1207535
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Abstract: Every real-valued convex and locally Lipschitzian function f defined on a nonempty closed convex set D of a Banach space E is the local restriction of a convex Lipschitzian function defined on E. Moreover, if E is separable and $ \operatorname{int} D \ne \emptyset $, then, for each Gateaux differentiability point x $ ( \in \operatorname{int} D)$ of f, there is a closed convex set $ C \subset \operatorname{int} D$ with the nonsupport points set $ N(C) \ne \emptyset $ and with $ x \in N(C)$ such that $ {f_C}$ (the restriction of f on C) is Fréchet differentiable at x.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1207535-X
PII: S 0002-9939(1994)1207535-X
Article copyright: © Copyright 1994 American Mathematical Society