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A function is defined in a code statement rather than declared with the usual function syntax. It has no name but is callable via a function reference. Such functions can be defined inside of a function as well as in other scopes. To use local variables in the anonymous function, use closure.
This is achieved by replacing multiple function definitions with a single function definition, which sets a list of variables equal to a list of expressions. A version of the Y combinator, called the Y* poly-variadic fix-point combinator [ 5 ] is then used to calculate fixed point of all the functions at the same time.
Anonymous functions are often arguments being passed to higher-order functions or used for constructing the result of a higher-order function that needs to return a function. [1] If the function is only used once, or a limited number of times, an anonymous function may be syntactically lighter than using a named function.
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
Currying provides a way for working with functions that take multiple arguments, and using them in frameworks where functions might take only one argument. For example, some analytical techniques can only be applied to functions with a single argument. Practical functions frequently take more arguments than this.
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), returns a function or value as its result. All other functions are first-order functions.
Multivalued functions of a complex variable have branch points. For example, for the nth root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units i and −i are branch points. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range.
Folds can be regarded as consistently replacing the structural components of a data structure with functions and values. Lists, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called nil ([]), or is constructed by prefixing an element in front of another list, creating what is called a cons node ( Cons(X1,Cons(X2,Cons ...