Functionem

Functionem is a theoretical construct used in mathematics and computer science to describe a parameterized family of functions together with its evaluation mechanism. In its simplest form, a functionem consists of an index set I and a family of functions {f_i : D -> C} indexed by i in I. The evaluation map Eval: I × D -> C is defined by Eval(i, x) = f_i(x). This view makes explicit that an input x is first paired with a choice of function i, after which the selected function is applied to x. If one fixes i, the corresponding function f_i is recovered by composing Eval with the inclusion {i} × D -> I × D.

Properties and variants: A functionem inherits properties from its component functions. If each f_i is continuous (between topological spaces D and C), then Eval is jointly continuous under suitable topologies. If the family is finite, functionems reduce to ordinary finite families of functions. A common operation is parameter-wise composition: for a fixed g: C -> E, the family {g ∘ f_i} forms a new functionem with the same index set.

Examples: The family f_theta(t) = sin(t + theta) for theta in [0, 2π) yields Eval(theta, t) = sin(t + theta). In programming, a functionem corresponds to a function that takes a pair (parameter, input) and returns an output.

Extensions: In a dependent variant, the codomain can depend on i, giving a map Eval: I × D -> C_i.

See also: function, parameterized function, evaluation map, indexed family.