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*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>\arg (a+bi) + 2 \pi n i</math> for all [[integer]]s <math>n</math>.
*[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have
▲*[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have
::<math>
\tan\left({\textstyle\frac{\pi}{4}}\right) = \tan\left({\textstyle\frac{5\pi}{4}}\right)
= \tan\left({\textstyle\frac{-3\pi}{4}}\right) = \tan\left({\textstyle\frac{(2n+1)\pi}{4}}\right) = \cdots = 1.
</math>
:As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the ___domain of tan ''x'' to -π/2 < ''x'' < π/2 – a ___domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes -π/2 < ''y'' < π/2. These values from a restricted ___domain are called ''[[principal value]]s''.
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==Applications==
Multifunctions arise in [[Optimal control|optimal control theory]], especially [[differential inclusion]]s and related subjects as [[game theory]], where the [[Kakutani fixed point theorem]] for multifunctions has been applied to prove existence of [[Nash equilibrium|Nash equilibria]] (note: in the context of game theory, a multivalued function is usually referred to as a [[correspondence (mathematics)|correspondence]].) This among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.
Nevertheless, lower hemicontinuous multifunctions usually possess continuous selections as stated in the [[Michael selection theorem]], which provides another characterisation of [[paracompact]] spaces (see: E. Michael, Continuous selections I" Ann. of Math. (2) 63 (1956), and D. Repovs, P.V. Semenov, Ernest Michael and theory of continuous selections" arXiv:0803.4473v1). Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski—Ryll-Nardzewski measurable selection, Aumann measurable selection, Fryszkowski selection for decomposable maps are important in [[optimal control]] and the theory of [[differential inclusion]]s.
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