Functional Programming

3.1. First-class and higher-order functions

Higher-order functions are functions that can either take other functions as arguments or return them as results. In calculus, an example of a higher-order function is the differential operator {\displaystyle d/dx}d/dx , which returns the derivative of a function {\displaystyle f}f.

Higher-order functions are closely related to first-class functions in that higher-order functions and first-class functions both allow functions as arguments and results of other functions. The distinction between the two is subtle: "higher-order" describes a mathematical concept of functions that operate on other functions, while "first-class" is a computer science term for programming language entities that have no restriction on their use (thus first-class functions can appear anywhere in the program that other first-class entities like numbers can, including as arguments to other functions and as their return values).

Higher-order functions enable partial application or currying, a technique that applies a function to its arguments one at a time, with each application returning a new function that accepts the next argument. This lets a programmer succinctly express, for example, the successor function as the addition operator partially applied to the natural number one.