Quasicontinuous functions, domains, and extended calculus

Abstract

One of the aims of domain theory is the construction of an embedding of a given structure or data type as the maximal or “ideal” elements of an enveloping domain of “approximations,” sometimes called a domain environment. Typically the goal is to provide a computational model or framework for recursive and algorithmic reasoning about the original structure. In this paper we consider the function space of (natural equivalence classes of) quasicontinuous functions from a locally compact space X into L, an n-fold product of the extended reals [−1,1] (more generally, into a bicontinuous lattice). We show that the domain of all “approximate maps” that assign to each point of X an order interval of L is a domain environment for the quasicontinuous function space. We rely upon the theory of domain environments to introduce an interesting and useful function space topology on the quasicontinuous function space. We then apply this machinery to define an extended differential calculus in the quasicontinuous function space, and draw connections with viscosity solutions of Hamiltonian equations. The theory depends heavily on topological properties of quasicontinuous functions that have been recently uncovered that involve dense sets of points of continuity and sections of closed relations and USCO maps. These and other basic results about quasicontinuous functions are surveyed and presented in the early sections.