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<title>Multivalued function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Multivalued function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about multivalued functions as they are considered in mathematical analysis. For set-valued functions as considered in variational analysis, see <a href="Set-valued_function" title="Set-valued function">set-valued function</a>.</div><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Multivariate_function" class="mw-redirect" title="Multivariate function">Multivariate function</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>multivalued function</b>,<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> <b>multiple-valued function</b>,<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> <b>many-valued function</b>,<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> or <b>multifunction</b>,<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> is a function that has two or more values in its range for at least one point in its domain.<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> It is a <a href="Set-valued_function" title="Set-valued function">set-valued function</a> with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions,<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> but English Wikipedia currently does, having a separate article for each.

</p><p>A <i>multivalued function</i> of sets <i>f : X → Y</i> is a subset

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.}</annotation>

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</math></span><img src="./ed0d3f57edcd92d1c87348d3e4b31a1f22522903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.089ex; height:2.843ex;" alt="{\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.}" loading="lazy"></span></dd></dl>

<p>Write <i>f(x)</i> for the set of those <i>y</i> ∈ <i>Y</i> with (<i>x,y</i>) ∈ <i>Γ<sub>f</sub></i>. If <i>f</i> is an ordinary function, it is a multivalued function by taking its <a href="Graph_of_a_function" title="Graph of a function">graph</a>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.}</annotation>

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</math></span><img src="./35de8599effa0e7b603b0c7b183a36180647036e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.66ex; height:3.009ex;" alt="{\displaystyle \Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.}" loading="lazy"></span></dd></dl>

<p>They are called <b>single-valued functions</b> to distinguish them.

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<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Global_analytic_function" title="Global analytic function">Global analytic function</a></div>

<p>The term multivalued function originated in complex analysis, from <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a>. It often occurs that one knows the value of a complex <a href="Analytic_function" title="Analytic function">analytic function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation>

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</math></span><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"></span> in some <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of a point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a}">

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<annotation encoding="application/x-tex">{\displaystyle z=a}</annotation>

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</math></span><img src="./fe11284406a9c3d677955fd62cf6ead90ae2070d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.416ex; height:1.676ex;" alt="{\displaystyle z=a}" loading="lazy"></span>. This is the case for functions defined by the <a href="Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> or by a <a href="Taylor_series" title="Taylor series">Taylor series</a> around <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a}">

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<annotation encoding="application/x-tex">{\displaystyle z=a}</annotation>

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</math></span><img src="./fe11284406a9c3d677955fd62cf6ead90ae2070d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.416ex; height:1.676ex;" alt="{\displaystyle z=a}" loading="lazy"></span>. In such a situation, one may extend the domain of the single-valued function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation>

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</math></span><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"></span> along curves in the complex plane starting at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>. In doing so, one finds that the value of the extended function at a point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=b}">

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<annotation encoding="application/x-tex">{\displaystyle z=b}</annotation>

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</math></span><img src="./b655a5698d722586f63fc8062ef3206439685eb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.184ex; height:2.176ex;" alt="{\displaystyle z=b}" loading="lazy"></span> depends on the chosen curve from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span>; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.

</p><p>For example, let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\sqrt {z}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\sqrt {z}}\,}</annotation>

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</math></span><img src="./54ea99a097e314a5ba5c4f5ac8b24b44dd32084f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.686ex; height:3.009ex;" alt="{\displaystyle f(z)={\sqrt {z}}\,}" loading="lazy"></span> be the usual <a href="Square_root" title="Square root">square root</a> function on positive real numbers. One may extend its domain to a neighbourhood of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}">

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<annotation encoding="application/x-tex">{\displaystyle z=1}</annotation>

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</math></span><img src="./078535cde78d90bfa1d9fbb2446204593a921d57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}" loading="lazy"></span> in the complex plane, and then further along curves starting at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}">

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<annotation encoding="application/x-tex">{\displaystyle z=1}</annotation>

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</math></span><img src="./078535cde78d90bfa1d9fbb2446204593a921d57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}" loading="lazy"></span>, so that the values along a given curve vary continuously from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1}}=1}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {1}}=1}</annotation>

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</math></span><img src="./5509eac62c3da5cefd034b3ca1f2b2f5f2e9c3ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1}}=1}" loading="lazy"></span>. Extending to negative real numbers, one gets two opposite values for the square root—for example <span class="texhtml">±<i>i</i></span> for <span class="texhtml">−1</span>—depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for <a href="Nth_root" title="Nth root"><span class="texhtml mvar" style="font-style:italic;">n</span>th roots</a>, <a href="Logarithm" title="Logarithm">logarithms</a>, and <a href="Inverse_trigonometric_function" class="mw-redirect" title="Inverse trigonometric function">inverse trigonometric functions</a>.

</p><p>To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the <a href="Principal_value" title="Principal value">principal value</a>, producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path (<a href="Monodromy" title="Monodromy">monodromy</a>). These problems are resolved in the theory of <a href="Riemann_surface" title="Riemann surface">Riemann surfaces</a>: to consider a multivalued function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation>

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</math></span><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"></span> as an ordinary function without discarding any values, one multiplies the domain into a many-layered <a href="Branched_covering" title="Branched covering">covering space</a>, a <a href="Manifold" title="Manifold">manifold</a> which is the Riemann surface associated to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation>

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</math></span><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Inverses_of_functions">Inverses of functions</h2></div>

<p>If <i>f : X → Y</i> is an ordinary function, then its inverse is the multivalued function

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{f^{-1}}\ \subseteq \ Y\times X}</annotation>

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</math></span><img src="./1899dcacf086a4637948cf046d4c80712f416f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.342ex; height:3.176ex;" alt="{\displaystyle \Gamma _{f^{-1}}\ \subseteq \ Y\times X}" loading="lazy"></span></dd></dl>

<p>defined as <i>Γ<sub>f</sub></i>, viewed as a subset of <i>X</i> × <i>Y</i>. When <i>f</i> is a <a href="Differentiable_function" title="Differentiable function">differentiable function</a> between <a href="Manifold" title="Manifold">manifolds</a>, the <a href="Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a> gives conditions for this to be single-valued locally in <i>X</i>.

</p><p>For example, the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> <i>log(z)</i> is the multivalued inverse of the exponential function <i>e<sup>z</sup></i> : <b>C</b> → <b>C</b><sup>×</sup>, with graph

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\log(z)}\ =\ \{(z,w)\ :\ w=\log(z)\}\ \subseteq \ \mathbf {C} \times \mathbf {C} ^{\times }.}</annotation>

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</math></span><img src="./562534e1f103111685c18149ec1ce6bea668e4de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.866ex; height:3.176ex;" alt="{\displaystyle \Gamma _{\log(z)}\ =\ \{(z,w)\ :\ w=\log(z)\}\ \subseteq \ \mathbf {C} \times \mathbf {C} ^{\times }.}" loading="lazy"></span></dd></dl>

<p>It is not single valued, given a single <i>w</i> with <i>w = log(z)</i>, we have

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<annotation encoding="application/x-tex">{\displaystyle \log(z)\ =\ w\ +\ 2\pi i\mathbf {Z} .}</annotation>

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</math></span><img src="./936fca46c88abf26ca7c05468d0af6949fea1b94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.372ex; height:2.843ex;" alt="{\displaystyle \log(z)\ =\ w\ +\ 2\pi i\mathbf {Z} .}" loading="lazy"></span></dd></dl>

<p>Given any <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> function on an open subset of the <a href="Complex_plane" title="Complex plane">complex plane</a> <b>C</b>, its <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> is always a multivalued function.

</p>

<div class="mw-heading mw-heading2"><h2 id="Concrete_examples">Concrete examples</h2></div>

<ul><li>Every <a href="Real_number" title="Real number">real number</a> greater than zero has two real <a href="Square_root" title="Square root">square roots</a>, so that square root may be considered a multivalued function. For example, we may write <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {4}}=\pm 2=\{2,-2\}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {4}}=\pm 2=\{2,-2\}}</annotation>

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</math></span><img src="./c222c68c1780a8346cb75e9984e6c4c50703dfde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.758ex; height:3.176ex;" alt="{\displaystyle {\sqrt {4}}=\pm 2=\{2,-2\}}" loading="lazy"></span>; although zero has only one square root, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {0}}=\{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {0}}=\{0\}}</annotation>

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</math></span><img src="./15e3fd915f5bb541593e0a22cdc36825eccfc9be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.684ex; height:3.009ex;" alt="{\displaystyle {\sqrt {0}}=\{0\}}" loading="lazy"></span>. Note that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}}</annotation>

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</math></span><img src="./d62b24be305beff66cba9bfbcc01a362ba390f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}}" loading="lazy"></span> usually denotes only the principal square root of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>.</li>

<li>Each nonzero <a href="Complex_number" title="Complex number">complex number</a> has two square roots, three <a href="Cube_root" title="Cube root">cube roots</a>, and in general <i>n</i> <a href="Nth_root" title="Nth root"><i>n</i>th roots</a>. The only <i>n</i>th root of 0 is 0.</li>

<li>The <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> function is multiple-valued. The values assumed by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(a+bi)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log(a+bi)}</annotation>

</semantics>

</math></span><img src="./9bbacc5f90840d5b86428ab3bddf904cccabac5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.651ex; height:2.843ex;" alt="{\displaystyle \log(a+bi)}" loading="lazy"></span> for real numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log {\sqrt {a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log {\sqrt {a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni}</annotation>

</semantics>

</math></span><img src="./498fe02663d5e5a3df8c2a29662b640c491c37ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.337ex; height:3.509ex;" alt="{\displaystyle \log {\sqrt {a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni}" loading="lazy"></span> for all <a href="Integer" title="Integer">integers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>.</li>

<li><a href="Inverse_trigonometric_function" class="mw-redirect" title="Inverse trigonometric function">Inverse trigonometric functions</a> are multiple-valued because trigonometric functions are periodic. We have <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left({\tfrac {\pi }{4}}\right)=\tan \left({\tfrac {5\pi }{4}}\right)=\tan \left({\tfrac {-3\pi }{4}}\right)=\tan \left({\tfrac {(2n+1)\pi }{4}}\right)=\cdots =1.}">

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \left({\tfrac {\pi }{4}}\right)=\tan \left({\tfrac {5\pi }{4}}\right)=\tan \left({\tfrac {-3\pi }{4}}\right)=\tan \left({\tfrac {(2n+1)\pi }{4}}\right)=\cdots =1.}</annotation>

</semantics>

</math></span></span> As a consequence, arctan(1) is intuitively related to several values: <span class="texhtml mvar" style="font-style:italic;">π</span>/4, 5<span class="texhtml mvar" style="font-style:italic;">π</span>/4, −3<span class="texhtml mvar" style="font-style:italic;">π</span>/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan <i>x</i> to <span class="nowrap">−<span class="texhtml mvar" style="font-style:italic;">π</span>/2 < <i>x</i> < <span class="texhtml mvar" style="font-style:italic;">π</span>/2</span> – a domain over which tan <i>x</i> is monotonically increasing. Thus, the range of arctan(<i>x</i>) becomes <span class="nowrap">−<span class="texhtml mvar" style="font-style:italic;">π</span>/2 < <i>y</i> < <span class="texhtml mvar" style="font-style:italic;">π</span>/2</span>. These values from a restricted domain are called <i><a href="Principal_value" title="Principal value">principal values</a></i>.</li>

<li>The <a href="Antiderivative" title="Antiderivative">antiderivative</a> can be considered as a multivalued function. The antiderivative of a function is the set of functions whose derivative is that function. The <a href="Constant_of_integration" title="Constant of integration">constant of integration</a> follows from the fact that the derivative of a constant function is 0.</li>

<li><a href="Inverse_hyperbolic_functions" title="Inverse hyperbolic functions">Inverse hyperbolic functions</a> over the complex domain are multiple-valued because hyperbolic functions are periodic along the imaginary axis. Over the reals, they are single-valued, except for arcosh and arsech.</li></ul>

<p>These are all examples of multivalued functions that come about from non-<a href="Injective_function" title="Injective function">injective functions</a>. Since the original functions do not preserve all the information of their inputs, they are not reversible. Often, the restriction of a multivalued function is a <a href="Partial_inverse" class="mw-redirect" title="Partial inverse">partial inverse</a> of the original function.

</p>

<div class="mw-heading mw-heading2"><h2 id="Branch_points">Branch points</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Branch_point" title="Branch point">Branch point</a></div>

<p>Multivalued functions of a complex variable have <a href="Branch_point" title="Branch point">branch points</a>. For example, for the <i>n</i>th root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units <i>i</i> and −<i>i</i> are branch points. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a>, a kind of curve that connects pairs of branch points, thus reducing the multilayered <a href="Riemann_surface" title="Riemann surface">Riemann surface</a> of the function to a single layer. As in the case with real functions, the restricted range may be called the <i>principal branch</i> of the function.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>In physics, multivalued functions play an increasingly important role. They form the mathematical basis for <a href="Paul_Dirac" title="Paul Dirac">Dirac</a>'s <a href="Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a>, for the theory of <a href="Crystallographic_defect" title="Crystallographic defect">defects</a> in crystals and the resulting <a href="Plasticity_(physics)" title="Plasticity (physics)">plasticity</a> of materials, for <a href="Vortex" title="Vortex">vortices</a> in <a href="Superfluid" class="mw-redirect" title="Superfluid">superfluids</a> and <a href="Superconductor" class="mw-redirect" title="Superconductor">superconductors</a>, and for <a href="Phase_transition" title="Phase transition">phase transitions</a> in these systems, for instance <a href="Melting" title="Melting">melting</a> and <a href="Quark_confinement" class="mw-redirect" title="Quark confinement">quark confinement</a>. They are the origin of <a href="Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a> structures in many branches of physics.

</p>

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation (mathematics)</a></li>

<li><a href="Function_(mathematics)" title="Function (mathematics)">Function (mathematics)</a></li>

<li><a href="Binary_relation" title="Binary relation">Binary relation</a></li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a href="Hagen_Kleinert" title="Hagen Kleinert">H. Kleinert</a>, <i>Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation</i>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080315225354/http://www.worldscibooks.com/physics/6742.html">World Scientific (Singapore, 2008)</a> (also available <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/re.html#B9">online</a>)</li>

<li><a href="Hagen_Kleinert" title="Hagen Kleinert">H. Kleinert</a>, <i>Gauge Fields in Condensed Matter</i>, Vol. I: Superflow and Vortex Lines, 1–742, Vol. II: Stresses and Defects, 743–1456, World Scientific, Singapore, 1989 (also available online: <a rel="nofollow" class="external text" href="http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html">Vol. I</a> and <a rel="nofollow" class="external text" href="http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html">Vol. II</a>)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.lib.msu.edu/crcmath/math/math/m/m450.htm">"Multivalued Function"</a>. <i>archive.lib.msu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-25</span></span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-fall-1999/pages/study-materials/multiple-valued-functions/">"Multiple Valued Functions | Complex Variables with Applications | Mathematics"</a>. <i>MIT OpenCourseWare</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-25</span></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFAl-RabadiZwick2004" class="citation journal cs1">Al-Rabadi, Anas; Zwick, Martin (2004-01-01). <a rel="nofollow" class="external text" href="https://pdxscholar.library.pdx.edu/sysc_fac/30/">"Modified Reconstructability Analysis for Many-Valued Functions and Relations"</a>. <i>Kybernetes</i>. <b>33</b> (5/6): <span class="nowrap">906–</span>920. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1108%2F03684920410533967">10.1108/03684920410533967</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFLedyaevZhu1999" class="citation journal cs1">Ledyaev, Yuri; Zhu, Qiji (1999-09-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://scholarworks.wmich.edu/math_pubs/22/">"Implicit Multifunction Theorems"</a></span>. <i>Set-Valued Analysis Volume</i>. <b>7</b> (3): <span class="nowrap">209–</span>238. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1008775413250">10.1023/A:1008775413250</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MultivaluedFunction.html">"Multivalued Function"</a>. <i>Wolfram MathWorld</i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 February</span> 2024</span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFRepovš1998" class="citation book cs1">Repovš, Dušan (1998). <i>Continuous selections of multivalued mappings</i>. Pavel Vladimirovič. Semenov. Dordrecht: Kluwer Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-5277-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/39739641">39739641</a>.</cite></span>

</li>

</ol></div></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">

</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="Identity_function" title="Identity function">Identity</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="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="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 2025-08-06" href="https://en.wikipedia.org/wiki/?title=Multivalued_function&oldid=1304504981">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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