Set-Valued Analysis

, Volume 4, Issue 4, pp 375–398 | Cite as

Approximate subgradients and coderivatives in Rn

  • D. Borwein
  • J. M. Borwein
  • Xianfu Wang
Article

Abstract

We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.

Mathematics Subject Classifications (1991)

Primary 49J52 Secondary 26A27 26B12 49J50 52A20 

Key words

subgradient coderivative generalized Jacobian Lipschitz function bump function gauge nowhere dense set Lebesgue measure disconnectedness 

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Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • D. Borwein
    • 1
  • J. M. Borwein
    • 2
  • Xianfu Wang
    • 3
  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada
  3. 3.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada

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