Functionwise

Functionwise is a term encountered in mathematics and computer science to describe operations, properties, or constructions that are applied to functions as objects, rather than to the inputs or outputs of those functions.

In mathematics, a functionwise operation takes a set or family of functions and transforms each member independently. For example, given a family {f_i}, a functionwise map T yields {T(f_i)}. This concept is often contrasted with pointwise operations, which act on the output values f_i(x) for each x. Informal examples include functionwise differentiation (applying a derivative to each function in a collection to obtain {f_i'}), or functionwise composition (forming a new family by composing each f_i with a fixed function).

In programming, functionwise usage describes higher-order constructs that operate on functions themselves. A functionwise map can apply a transformation to a collection of functions, returning a new collection, or a functionwise combinator can combine multiple function-valued arguments into a single function. Functional programming languages frequently provide utilities for mapping, composing, or applying over function-valued data, reflecting the idea of manipulating functions as first-class values.

Because functionwise is not a rigid, universally defined term, its meaning depends on context. It is often used informally to emphasize that an operation targets functions rather than their inputs or outputs. Related concepts include higher-order functions, pointwise operations, and functional programming.