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A Function that takes as an argument or returns a function

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A higher-order function is a function that takes functions as arguments or returns functions as results. Consider the following Language:Haskell function which applies a given argument function twice:

twice :: (x -> x) -> x -> x
twice f = f . f

For instance:

> twice (+1) 40
42

twice is clearly a higher-order function in that its first argument is of a function type "x -> x". twice is actually also a higher-order function in that it composes a function that it returns as result, namely "f . f". The function composition operator "." is another higher-order function, which is obvious from its type, again:

> :t (.)
(.) :: (b -> c) -> (a -> b) -> a -> c

For what it matters, the type itself already reveals that the second function is applied before the first function because of how type "a" of the third argument equals with the argument type of the second function, etc. Now it would be straightforward to define function composition (except that we do not need to do, as it is predefined evidently):

(.) f g x = f (g x)

See list processing functions such as map, fold, and filter for more examples of higher-order functions. See the concept of currying as well.

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