limitedfunction

Limitedfunction is a mathematical concept referring to a function whose range is finite. Formally, a function f: X -> Y is called a limitedfunction when the set {f(x) : x in X} is finite. The term finite-range function is used interchangeably by some authors.

Examples include constant functions, which have a range consisting of a single value; indicator or characteristic functions of a subset A of X, which take only two values; and modular mappings such as f(x) = x mod n, which map the domain to a finite set {0,1,...,n-1}. Even nontrivial real-valued functions can be limited on restricted domains; for instance, f: R -> R with f(x) = sin x has range [-1,1], which is not finite, but f(x) = sign(sin x) has range { -1, 0, 1 } and is thus a limitedfunction.

Properties of limitedfunctions include that they partition their domain into at most |Y| preimage sets, one for each value in the range. The finiteness of the image implies certain closure properties under composition and arithmetic: if g: Y -> Z is any function, then g∘f has finite image, since its values lie in g(f(X)) which is a finite set. Likewise, if f and h: X -> R have finite images A and B, then f+h and f·h have images contained in A+B and A·B, respectively; both are finite.

Applications of the concept appear in discrete mathematics and computer science as a way to model finite-state-like behavior or discretized outputs. Finite-range functions arise naturally in data encoding, hashing with limited output alphabets, and stepwise approximations of continuous phenomena. See also finite-range function; finite-state machine; step function.