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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Identity function</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Null_function" title="Null function">Null function</a> or <a href="Empty_function" class="mw-redirect" title="Empty function">Empty function</a>.</div>
<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>identity function</b>, also called an <b>identity relation</b>, <b>identity map</b> or <b>identity transformation</b>, is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that always returns the value that was used as its <a href="Argument_of_a_function" title="Argument of a function">argument</a>, unchanged. That is, when <span class="texhtml mvar" style="font-style:italic;">f</span> is the identity function, the <a href="Equality_(mathematics)" title="Equality (mathematics)">equality</a> <span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i></span> is true for all values of <span class="texhtml mvar" style="font-style:italic;">x</span> to which <span class="texhtml mvar" style="font-style:italic;">f</span> can be applied.
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Formally, if <span class="texhtml"><i>X</i></span> is a <a href="Set_(mathematics)" title="Set (mathematics)">set</a>, the identity function <span class="texhtml"><i>f</i></span> on <span class="texhtml"><i>X</i></span> is defined to be a function with <span class="texhtml"><i>X</i></span> as its <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a>, satisfying
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</style><div class="block-indent" style="padding-left: 1.6em;"><span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i></span> for all elements <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>X</i></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></div>
<p>In other words, the function value <span class="texhtml"><i>f</i>(<i>x</i>)</span> in the codomain <span class="texhtml"><i>X</i></span> is always the same as the input element <span class="texhtml"><i>x</i></span> in the domain <span class="texhtml"><i>X</i></span>. The identity function on <span class="texhtml mvar" style="font-style:italic;">X</span> is clearly an <a href="Injective_function" title="Injective function">injective function</a> as well as a <a href="Surjective_function" title="Surjective function">surjective function</a> (its codomain is also its <a href="Range_(function)" class="mw-redirect" title="Range (function)">range</a>), so it is <a href="Bijection" title="Bijection">bijective</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>The identity function <span class="texhtml"><i>f</i></span> on <span class="texhtml"><i>X</i></span> is often denoted by <span class="texhtml">id<sub><i>X</i></sub></span>.
</p><p>In <a href="Set_theory" title="Set theory">set theory</a>, where a function is defined as a particular kind of <a href="Binary_relation" title="Binary relation">binary relation</a>, the identity function is given by the <a href="Identity_relation" class="mw-redirect" title="Identity relation">identity relation</a>, or <i>diagonal</i> of <span class="texhtml"><i>X</i></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Algebraic_properties">Algebraic properties</h2></div>
<p>If <span class="texhtml"><i>f</i> : <i>X</i> → <i>Y</i></span> is any function, then <span class="texhtml"><i>f</i> ∘ id<sub><i>X</i></sub> = <i>f</i> = id<sub><i>Y</i></sub> ∘ <i>f</i></span>, where "∘" denotes <a href="Function_composition" title="Function composition">function composition</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In particular, <span class="texhtml">id<sub><i>X</i></sub></span> is the <a href="Identity_element" title="Identity element">identity element</a> of the <a href="Monoid" title="Monoid">monoid</a> of all functions from <span class="texhtml"><i>X</i></span> to <span class="texhtml"><i>X</i></span> (under function composition).
</p><p>Since the identity element of a monoid is <a href="Unique_(mathematics)" class="mw-redirect" title="Unique (mathematics)">unique</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> one can alternately define the identity function on <span class="texhtml"><i>M</i></span> to be this identity element. Such a definition generalizes to the concept of an <a href="Identity_morphism" class="mw-redirect" title="Identity morphism">identity morphism</a> in <a href="Category_theory" title="Category theory">category theory</a>, where the <a href="Endomorphism" title="Endomorphism">endomorphisms</a> of <span class="texhtml"><i>M</i></span> need not be functions.
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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<ul><li>The identity function is a <a href="Linear_map" title="Linear map">linear operator</a> when applied to <a href="Vector_space" title="Vector space">vector spaces</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>
<li>In an <span class="texhtml mvar" style="font-style:italic;">n</span>-<a href="Dimension_(vector_space)" title="Dimension (vector space)">dimensional</a> <a href="Vector_space" title="Vector space">vector space</a> the identity function is represented by the <a href="Identity_matrix" title="Identity matrix">identity matrix</a> <span class="texhtml"><i>I</i><sub><i>n</i></sub></span>, regardless of the <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> chosen for the space.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>
<li>The identity function on the positive <a href="Integer" title="Integer">integers</a> is a <a href="Completely_multiplicative_function" title="Completely multiplicative function">completely multiplicative function</a> (essentially multiplication by 1), considered in <a href="Number_theory" title="Number theory">number theory</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li>
<li>In a <a href="Metric_space" title="Metric space">metric space</a> the identity function is trivially an <a href="Isometry" title="Isometry">isometry</a>. An object without any <a href="Symmetry" title="Symmetry">symmetry</a> has as its <a href="Symmetry_group" title="Symmetry group">symmetry group</a> the <a href="Trivial_group" title="Trivial group">trivial group</a> containing only this isometry (symmetry type <span class="texhtml">C<sub>1</sub></span>).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li>
<li>In a <a href="Topological_space" title="Topological space">topological space</a>, the identity function is always <a href="Continuous_function#Continuous_functions_between_topological_spaces" title="Continuous function">continuous</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li>
<li>The identity function is <a href="Idempotence" title="Idempotence">idempotent</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Identity_matrix" title="Identity matrix">Identity matrix</a></li>
<li><a href="Inclusion_map" title="Inclusion map">Inclusion map</a></li>
<li><a href="Indicator_function" title="Indicator function">Indicator function</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><div id="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>
<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Boolean-valued_function" title="Boolean-valued function"><span class="texhtml">X → 𝔹</span></a></li>
<li><a href="Ordered_pair" title="Ordered pair"><span class="texhtml">𝔹 → X</span></a></li>
<li><a href="Boolean_function" title="Boolean function"><span class="texhtml">𝔹ⁿ → X</span></a></li>
<li><a href="Integer-valued_function" title="Integer-valued function"><span class="texhtml">X → ℤ</span></a></li>
<li><a href="Sequence" title="Sequence"><span class="texhtml">ℤ → X</span></a></li>
<li><a href="Real-valued_function" title="Real-valued function"><span class="texhtml">X → ℝ</span></a></li>
<li><a href="Function_of_a_real_variable" title="Function of a real variable"><span class="texhtml">ℝ → X</span></a></li>
<li><a href="Function_of_several_real_variables" title="Function of several real variables"><span class="texhtml">ℝⁿ → X</span></a></li>
<li><a href="Complex-valued_function" class="mw-redirect" title="Complex-valued function"><span class="texhtml">X → ℂ</span></a></li>
<li><a href="Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable"><span class="texhtml">ℂ → X</span></a></li>
<li><a href="Function_of_several_complex_variables" title="Function of several complex variables"><span class="texhtml">ℂⁿ → X</span></a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>
<li><a href="Linear_map" title="Linear map">Linear</a></li>
<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>
<li><a href="Rational_function" title="Rational function">Rational</a></li>
<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>
<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>
<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>
<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>
<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>
<li><a href="Injective_function" title="Injective function">Injective</a></li>
<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>
<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>
<li><a href="Function_composition" title="Function composition">Composition</a></li>
<li><a href="Lambda_calculus" title="Lambda calculus">λ</a></li>
<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>
<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>
<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>
<li><a href="Partial_function" title="Partial function">Partial</a></li>
<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>
<li><a href="Function_space" title="Function space">Space</a></li>
<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>
<li><a href="Morphism" title="Morphism">Morphism</a></li>
<li><a href="Functor" title="Functor">Functor</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>
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This article is issued from <a class="external text" title="Last edited on 2025-07-02" href="https://en.wikipedia.org/wiki/?title=Identity_function&oldid=1298455023">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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