Title: A continuous derivative for real-valued functions

Abstract:

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.