A Continuous Derivative for Real-Valued Functions

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.