Domain theory, which was introduced by Dana Scott in the early 1970s, works on the principle of incorporating partial elements into denotational domains. The resulting structures are ordered by a natural notion of refinement or approximation, and they carry a topology that expresses the process of passing to a limit.

We introduce a domain-theoretic framework for differential calculus. We define the set of primitive maps as well as the derivative of an interval-valued Scott continuous function on the domain of intervals, and show that they are dually related, providing an extension of the classical duality of differentiation and integration as in the fundamental theorem of calculus. It is shown that, for locally Lipschitz functions of a real variable, the domain-theoretic derivative coincides with the Clarke's derivative. We then construct a domain for differentiable real-valued functions of a real variable by pairing consistent information about the function and information about its derivative. The set of classical $C^1$ functions, equipped with its $C^1$ norm, is embedded into the set of maximal elements of this countably based, bounded complete continuous domain. This domain also provides a model for the differential properties of piecewise $C^1$ functions, locally Lipschitz functions and more generally of all continuous functions. We prove that consistency of function information and derivative information is decidable on rational step functions, which shows that our domain can be given an effective structure. We thus obtain a data type for differential calculus. As an immediate application, we present a domain-theoretic formulation of Picard's theorem, which provides a data type for solving differential equations.

We show that real-number computations in the interval-domain environment are ‘inherently parallel’ in a precise mathematical sense. We do this by reducing computations of the weak parallel-or operation on the Sierpinski domain to computations of the addition operation on the interval domain.

This article surveys a variety of approaches to integration theory that have arisen in the context of continuous domain theory. The presentation is set in a broader context of ‘positive’ analysis, analysis where order-theoretic notions and notions of positivity play a key role.

We show that a measurement $\mu$ on a continuous dcpo $D$ extends to a measurement $\skew3\bar{\mu}$ on the convex powerdomain ${\mathbf C} D$ iff it is a Lebesgue measurement. In particular, $\ker\mu$ must be metrisable in its relative Scott topology. Moreover, the space $\ker\skew3\bar{\mu}$ in its relative Scott topology is homeomorphic to the Vietoris hyperspace of $\ker\mu$, that is, the space of non-empty compact subsets of $\ker\mu$ in its Vietoris topology – the topology induced by any Hausdorff metric. This enables one to show that Hutchinson's theorem holds for any finite set of contractions on a domain with a Lebesgue measurement. Finally, after resolving the existence question for Lebesgue measurements on countably based domains, we uncover the following relationship between classical analysis and domain theory: for an $\omega$-continuous dcpo $D$ with $\max(D)$ regular, the Vietoris hyperspace of $\max(D)$ embeds in $\max({\mathbf C} D)$ as the kernel of a measurement on ${\mathbf C} D$.

Every compact metric space $X$ is homeomorphically embedded in an $\omega$-algebraic domain $D$ as the set of minimal limit (that is, non-finite) elements. Moreover, $X$ is a retract of the set $L(D)$ of all limit elements of $D$. Such a domain $D$ can be chosen so that it has property M and finite-branching, and the height of $L(D)$ is equal to the small inductive dimension of $X$. We also show that the small inductive dimension of $L(D)$ as a topological space is equal to the height of $L(D)$ for domains with property M. These results give a characterisation of the dimension of a space $X$ as the minimal height of $L(D)$ in which $X$ is embedded as the set of minimal elements. The domain in which we embed an $n$-dimensional compact metric space $X$ ($n \leq \infinity$) has a concrete structure in that it consists of finite/infinite sequences in $\{0,1,\bot\}$ with at most $n$ copies of $\bot$.