Extensions of Domain Maps in Differential and Integral Calculus

We introduce in the context of differential and integral calculus several key extensions of higher order maps from a dense subset of a topological space into a continuous Scott domain. These higher order maps include the classical derivative operator and the Riemann integration operator. Using a sequence of test functions, we prove that the subspace of real-valued continuously differentiable functions on a finite dimensional Euclidean space is dense in the space of Lipschitz maps equipped with the L-topology. This provides a new result in basic mathematical analysis, which characterises the L-topology in terms of the limsup of the sequence of derivatives of a sequence of C1 maps that converges to a Lipschitz map. Using this result, it is also shown that the generalised (Clarke) gradient on Lipschitz maps is the extension of the derivative operator on C1 maps. We show that the generalised Riemann integral (R-integral) of a real-valued continuous function on a compact metric space with respect to a Borel measure can be extended to the integral of interval-valued functions on the metric space with respect to valuations on the probabilistic power domain of the space of non-empty and compact sets of the metric space. We also prove that the Lebesgue integral operator on integrable functions is the extension of the R-integral operator on continuous functions. We finally illustrate an application of these results by deriving a simple proof of Green's theorem for interval-valued vector fields.

I. Introduction

The notion of injective spaces and densely injective spaces were introduced by Dana Scott in [30]. A topological space is said to be (densely) injective if every continuous map extends continuously to any pace containing as a (dense) subset. A celebrated result in domain theory is that a space is (densely) injective iff it is a continuous lattice (a bounded complete continuous domain) equipped with its Scott topology [20, p. 181–182]. This provides a topological characterisation of continuous lattices and bounded complete continuous dcop's, also referred to as bounded complete domains or as continuous Scott domains.