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. As with the Clarke gradient, the values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. The L-derivative, however, is shown to be upper semi continuous, a result which is not known to hold for the Clarke gradient. 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 notion of effectivity. For real-valued functions on finite dimensional Euclidean spaces, the L-derivative can be obtained within an effectively given continuous domain. We also show that in finite dimensions the L-derivative and the Clarke gradient coincide thus providing a computable representation for the latter in this case.This paper is dedicated to the historical memory of Sharaf al-din Tusi (d. 1213), the Iranian mathematician who was the first to use the derivative systematically to solve for roots of cubic polynomials and find their maxima.

A Language for Differentiable Functions

“…The L-derivative of the non-expansive map f : I → I is the Scott continuous function L(f ) : I → I defined by [5]:…”

“…The L-derivative was later defined and studied for real-valued functions on Banach spaces and it was shown that on finite dimensional Euclidean spaces the L-derivative actually coincides with the Clarke gradient [5]. In finite dimensions, therefore, the Lderivative provides a simple and finitary representation for the Clarke gradient, which in its original form was defined using an infinitary double limit superior operation.…”

Weak Topology and a Differentiable Operator for Lipschitz Maps

“…In [16], a domain-theoretic derivative was introduced for real-valued functions of the real line, which was later extended to higher dimensions [17,12] and shown to be mathematically equivalent to the Clarke gradient in finite dimensional spaces [12]. The L-derivative, as the domaintheoretic derivative is now called, has a number of distinct features compared with the Clarke gradient:…”

“…Any single step function of type U → C(R n ) defines a family of maps of type U → R as follows [12]. We say…”

“…It is shown in [12] that, in finite dimensional Euclidean spaces, the L-derivative coincides with the Clarke gradient [5]. It is however not known if they do coincide for infinite dimensional Banach spaces; see Section 2.2 below.…”