Intervalet

Intervalet is a term used in many European languages, including Norwegian, to denote an interval on the real number line. An interval is a connected, continuous range of real numbers bounded by two endpoints a and b, where a and b can be finite or infinite.

The basic, finite intervals are defined by endpoints a and b with a ≤ b. The closed interval [a,b] includes both endpoints. The open interval (a,b) excludes the endpoints. Half-open (or half-closed) intervals include one endpoint but not the other, namely [a,b) and (a,b].

Intervals can also be infinite. The closed left or right ray [a, ∞) or (−∞, b] extend without bound on one side, while the two-sided infinite interval (−∞, ∞) covers the entire real line. A degenerate interval occurs when a = b, containing a single point {a}.

In mathematics, intervals are fundamental because they describe domains of functions, ranges of values, and regions for solving inequalities. They are convex subsets of the real numbers, meaning any value between two points in the interval also lies in the interval. They serve as the building blocks for more complex sets and for concepts in calculus, analysis, and numerical methods. In higher dimensions, intervals generalize to axis-aligned boxes, such as [a1,b1] × [a2,b2], used in multivariable contexts and interval arithmetic for error-bounded computations.

The name intervalet traces its lineage to Latin intervallum, passing into modern languages through mathematical usage to denote a contiguous span of values.