Intervalldefinition

Intervalldefinition is a term encountered in some texts as a variant or misspelling of interval definition. In standard mathematical language, an interval is a connected subset of the real numbers described by two endpoints and an inclusion pattern. An interval definition specifies the endpoints and whether each endpoint is included. The basic types are closed intervals [a, b], open intervals (a, b), and half-open intervals [a, b) or (a, b]. Infinite intervals extend to infinity, such as [a, ∞) or (−∞, b]. The endpoints satisfy a ≤ b; if a = b with closed endpoints, the interval reduces to a single point {a}.

In set-builder terms, an interval I can be defined as {x in R : a ≤ x ≤ b} for [a, b], {x : a < x < b} for (a, b), and combinations for the half-open forms. Interval definitions are foundational in real analysis, topology, and numerical methods. They provide precise descriptions of domains, convergence regions, and constraint ranges. In computing, interval definitions appear in range specifications for programming languages, in interval arithmetic, and in database queries to bound values.

Common operations on interval definitions include union and intersection, which combine or narrow the applicable sets. When dealing with infinities or mixed endpoint inclusions, careful handling is required to preserve correctness. See also interval arithmetic, the real number line, and set notation.