Finiterange

Finite range, written as finiterange in some contexts, is a term used in mathematics and computer science to describe a mapping or variable whose image consists of only finitely many values. Formally, a function f: X -> Y has finite range if the set { f(x) : x ∈ X } is finite. If Y is a subset of the real numbers, this means f takes only a finite number of distinct values across its domain. A constant function is the simplest example, with range of size one. A function can have finite range even when its domain is infinite; conversely, if the domain is finite, the range is automatically finite.

In practice, finite range is often contrasted with the idea of a bounded range. A function can be defined on an infinite domain and have a bounded but infinite range (for example, values filling an interval). Finite range is thus a stronger condition: not only are values bounded or limited in extent, but there are only finitely many possible values.

Applications and related ideas:

- In computer science, the term is commonly used to describe the value set of a variable of fixed-width integer type. Such variables have a finite range determined by their bit width and signedness (for example, an 8-bit signed integer ranges from -128 to 127). This has implications for overflow and wrap-around behavior.

- In probability and statistics, a random variable has finite range (or finite support) when it takes only finitely many possible values.

- In mathematics, recognizing finite range can simplify enumeration, optimization, and proofs, since only finitely many outcomes require consideration.