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In mathematics, a function that always returns the same value that was used as its argument
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. That is, for f being identity, the equalityf(x) = x holds for all x.
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomainM which satisfies
In other words, the function value f(x) in M (that is, the codomain) is always the same input element x of M (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective.
The identity function f on M is often denoted by idM.
Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.