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Interval Domain: Simple Definition

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Generally speaking, an interval domain is a domain restricted to an interval [1]. For example, inputs (e.g. x-values) for a particular function might be restricted to the interval [0, 1]. Intervals can be closed, open, or half-closed/half-open.

interval domain

A function defined on the half open interval “interval domain” of [5.5, 10).


Interval Domain in Domain Theory

In domain theory, the term “interval domain”, first proposed by D.S. Scott in 1972 [2], is a way to approximate real numbers. It gets its name because the reals are divided into intervals for calculations.

Approximations are sometimes needed for calculations over uncountable spaces, such as the Reals (ℝ) or some function spaces. Interval domains are not as straightforward to define as the “intervals” you come across in calculus; Algebraic structures, which consist of a set plus one or more binary operations that to satisfy certain axioms, are needed to show the differences between the many equivalent (and non-equivalent) versions of interval domain [3].

Scott’s domain-theoretic framework for differential calculus was originally designed for single variable functions. It has more recently been extended to functions of several variables [3]. This extension carries the interval domain to approximations of curves and surfaces [4].

Domain theory and algebraic structures are beyond the scope of this article, but if you’re interested then read Jess Blanck’s Computer Journal article Interval Domains and Computable Sequences: A Case Study of Domain Reductions [5].


References

[1] Klippert, J. (1989). Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain. Mathematics Magazine
Vol. 62, No. 1 (Feb., 1989), pp. 43-48 (6 pages)
[2] Scott, D.S. (1972) Lattice Theory, Data Types and Semantics. In
Rustin, R. (ed.), Formal Semantics of Programming Languages,
pp. 65
[3] Edalat, A. (1995a) Domain theory and integration. Theoretical Computer Science 151 163–193.
[4] Edalat, A. & Lieutier, A. (2004). Domain theory and dfferential calculus(functions of one variable). Math. Struct. in Comp. Science, vol. 14, pp. 771–802. c2004 Cambridge University PressDOI: 10.1017/S0960129504004359 Printed in the United Kingdom
[5] Blanck, J. (2012). Interval Domains and Computable Sequences: A Case Study of Domain Reductions. The Computer Journal (Sep. 5).

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Stephanie Glen. "Interval Domain: Simple Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/interval-domain-definition/
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