Functionsor

Functionsor is a theoretical construct in mathematics and computer science describing a higher-order operator that merges multiple functions into a single output function via a selector mechanism. Given a family of functions {f_i: X -> Y} indexed by i in I and a selector map s: X -> I, the functionsor F is defined by F(x) = f_{s(x)}(x). It thus generalizes piecewise definitions and dynamic dispatch, allowing the choice of function to depend on the input.

Formally, F is well-defined whenever each f_i is defined on X and s(x) ∈ I for all x. Variants include probabilistic functionsor, where s selects i according to a probability distribution, yielding a stochastic evaluation, and stateful versions where the selector depends on internal state, enabling adaptive behavior.

An example: let f_0(x) = x^2 and f_1(x) = 2x + 1 on X = R, with s(x) = 0 for x < 0 and s(x) = 1 for x ≥ 0. Then F(x) = x^2 for x < 0 and F(x) = 2x + 1 for x ≥ 0, a simple piecewise function realized by functionsor.

Properties and connections: If all f_i are continuous and s is locally constant, F is piecewise continuous. If f_i are differentiable and s is constant on subdomains, F is differentiable on those subdomains. Functionsor relates to piecewise definitions, dispatch mechanisms in programming languages, and the study of higher-order compositions.

History: The term is used in hypothetical or exploratory discussions as a convenient label for function-dispatch constructions; it is not a standard mathematical operator. See also piecewise functions, conditional expressions, and multimethods.