A Continuous Derivative for Real-Valued Functions

  • Abbas Edalat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. The values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. This is also the case for the Clarke generalised gradient. The L-derivative, however, is shown to be upper semi continuous with respect to the weak* topology, a result which is not known to hold for the Clarke gradient on infinite dimensional Banach spaces. We also formulate the notion of primitive maps dual to the L-derivative, an extension of Fundamental Theorem of Calculus for the L-derivative and a domain for computation of real-valued functions on a Banach space with a corresponding computability theory.


L-derivative Clarke gradient weak* topology upper semi-continuity 


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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Abbas Edalat
    • 1
  1. 1.Department of Computing, Imperial College LondonUK

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