Intervalls

Intervalls are subsets of the real numbers that include all numbers between two endpoints. They are defined by two endpoints a and b with a ≤ b, though some interval concepts allow one or both endpoints to be infinite. The most common types are closed intervals [a,b], open intervals (a,b), and half-open intervals [a,b) or (a,b]. In a closed interval both endpoints are included; in an open interval neither endpoint is included; in half-open intervals exactly one endpoint is included.

Finite intervals have finite length b − a. A degenerate interval [a,a] contains a single point. The empty interval is the empty set, which can be represented in some conventions as an interval (a,b) with a > b, though it is sometimes treated as a separate case.

Notation uses brackets and parentheses to indicate inclusion. For example, [0,1] contains 0 and 1; (0,1) excludes both; [0,1) contains 0 but not 1. Unbounded intervals extend without bound: (a, ∞), (-∞, b], and the real line (-∞, ∞). These intervals are fundamental in expressing solution sets to inequalities, defining domains of functions, and specifying integration bounds. The concept extends to higher dimensions, where a multidimensional analogue is a box or hyperrectangle formed by the Cartesian product of intervals along each coordinate. Intervalls provide a simple, rigorous language for continuity, convergence, and many areas of analysis.