Funktion

Funktion, in English commonly called function, is a foundational concept in mathematics and related disciplines. In German-language mathematics, the term Funktion corresponds directly to the English term function; the concept is also central in computer science, statistics and logic. The word ultimately derives from Latin functio and French fonction.

A function f from a set X (the domain) to a set Y (the codomain) is a rule that assigns to every element x in X exactly one element y in Y, written f(x) = y. Equivalently, f is a subset of the Cartesian product X × Y with the property that each x ∈ X occurs with a single y ∈ Y. The domain of f is X, the codomain is Y, and the image or range is {f(x) : x ∈ X}.

Functions can be classified by properties such as injectivity, surjectivity and bijectivity. An injective function assigns distinct outputs to distinct inputs; a surjective function covers the entire codomain; a bijection is both. If f is bijective, it has an inverse f^−1. Function composition combines two functions, (g ∘ f)(x) = g(f(x)). Common types include linear, polynomial, exponential, logarithmic, and trigonometric functions, as well as piecewise definitions and constant functions. In analysis, notions of continuity, limits, and differentiability describe how a function behaves.

In programming, a function (or method) is a named block of code that maps inputs to outputs. Functions may be pure, with no side effects, or impure. In functional programming, functions are treated as first-class citizens.

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