functionn

FunctionN is a formal concept used in mathematics, computer science, and programming to denote a function that takes N arguments. In mathematics, an N-ary function f is a mapping from the Cartesian product A1 × A2 × ... × AN to B, written f: A1 × A2 × ... × AN → B. The inputs are an N-tuple (a1, a2, ..., aN) and the order of arguments matters.

Arity and currying: An N-ary function can be treated as a single function of N inputs or as a curried sequence of unary functions: f(a1)(a2)...(aN) = b. Currying reveals that, in many functional languages, functions with multiple arguments are retrievable by successive application of single-argument functions.

Partial application and composition: You can fix some arguments to obtain a function of fewer arguments, enabling partial application and function composition.

In programming languages and type systems, FunctionN is used as a generic notion for functions with N inputs. Some languages provide concrete Function1, Function2, etc., up to a fixed maximum arity, while others expose a general FunctionN form to represent arbitrary arity. Conceptually, FunctionN helps discuss topics such as arity, higher-order functions, and transformation of arguments.

Examples: The binary function f(a,b) = a + b maps integers to integers; a ternary function g(x,y,z) might compute a polynomial value g(x,y,z) = x*y + z.

See also currying, arity, higher-order functions, partial application.