Identity function

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title: "Identity function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["functions-and-mappings", "elementary-mathematics", "basic-concepts-in-set-theory", "types-of-functions", "1-(number)"] description: "Function that returns its argument unchanged" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Identity_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Function that returns its argument unchanged ::

thumb|[[Graph of a function|Graph]] of the identity function on the [[real number]]s

In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(x)=x is true for all values of x to which f can be applied.

Definition

Formally, if X is a set, the identity function f on X is defined to be a function with X as its domain and codomain, satisfying

In other words, the function value f(x) in the codomain X is always the same as the input element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.

The identity function f on X is often denoted by \mathrm{id}_X.

In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of X.

Algebraic properties

If f:X\rightarrow Y is any function, then f\circ\mathrm{id}_X=f=\mathrm{id}_Y\circ f, where "\circ" denotes function composition.{{cite book | last = Nel | first = Louis | year = 2016 | title = Continuity Theory | url = https://books.google.com/books?id=_JdPDAAAQBAJ&pg=PA21 | page = 21 | publisher = Springer | location = Cham | doi = 10.1007/978-3-319-31159-3 | isbn = 978-3-319-31159-3

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.

Properties

• The identity function is a linear operator when applied to vector spaces.
• In an n-dimensional vector space the identity function is represented by the identity matrix I_n, regardless of the basis chosen for the space.
• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
• In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type \mathrm{C}_1).
• In a topological space, the identity function is always continuous.
• The identity function is idempotent.

References

References

1. Knapp, Anthony W.. (2006). "Basic algebra". Springer.
2. Mapa, Sadhan Kumar. (7 April 2014). "Higher Algebra Abstract and Linear". Sarat Book House.
3. (1974). "Proceedings of Symposia in Pure Mathematics". American Mathematical Society.
4. (1999). "Finitely Generated Commutative Monoids". Nova Publishers.
5. Anton, Howard. (2005). "Elementary Linear Algebra (Applications Version)". Wiley International.
6. T. S. Shores. (2007). "Applied Linear Algebra and Matrix Analysis". Springer.
7. (2007). "Number Theory through Inquiry". Mathematical Assn of Amer.
8. Anderson, James W.. (2007). "Hyperbolic geometry". Springer.
9. Conover, Robert A.. (2014-05-21). "A First Course in Topology: An Introduction to Mathematical Thinking". Courier Corporation.
10. Conferences, University of Michigan Engineering Summer. (1968). "Foundations of Information Systems Engineering".

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