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Reverted 2 edits by Sheddow (talk): We are not in square root nor in real number |
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== Distinction from set-valued relations ==
[[File:Multivalued_functions_illustration.svg|thumb|right|
Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued relations (also called [[Set-valued function|set-valued functions]]) by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a [[Function (mathematics)|function]].<ref name=":0">{{Cite book |last1=Wriggers |first1=Peter |url=https://books.google.com/books?id=R4lqCQAAQBAJ |title=New Developments in Contact Problems |last2=Panatiotopoulos |first2=Panagiotis |date=2014-05-04 |publisher=Springer |isbn=978-3-7091-2496-3 |pages=29 |language=en}}</ref> Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.<ref name=":0" />
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==Concrete examples==
*Every [[real number]] greater than zero has two real [[square root]]s, so that square root may be considered a multivalued function. For example, we may write <math>\sqrt{4}=\pm 2=\{2,-2\}</math>; although zero has only one square root, <math>\sqrt{0} =\{0\}
*Each nonzero [[complex number]] has two square roots, three [[cube root]]s, and in general ''n'' [[nth root|''n''th roots]]. The only ''n''th root of 0 is 0.
*The [[complex logarithm]] function is multiple-valued. The values assumed by <math>\log(a+bi)</math> for real numbers <math>a</math> and <math>b</math> are <math>\log{\sqrt{a^2 + b^2}} + i\arg (a+bi) + 2 \pi n i</math> for all [[integer]]s <math>n</math>.
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