Editing Multivalued function (section)

==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\}</math>. Note that <math>\sqrt{x}</math> usually denotes only the principal square root of <math>x</math>.
*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>.
*[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have <math display="block">
\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.
</math> As a consequence, arctan(1) is intuitively related to several values: {{pi}}/4, 5{{pi}}/4, −3{{pi}}/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan ''x'' to {{nowrap|−{{pi}}/2 < ''x'' < {{pi}}/2}} – a domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes {{nowrap|−{{pi}}/2 < ''y'' < {{pi}}/2}}. These values from a restricted domain are called ''[[principal value]]s''.
* The [[antiderivative]] can be considered as a multivalued function. The antiderivative of a function is the set of functions whose derivative is that function. The [[constant of integration]] follows from the fact that the derivative of a constant function is 0.
*[[Inverse hyperbolic functions]] 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.

These are all examples of multivalued functions that come about from non-[[injective function]]s. 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 [[partial inverse]] of the original function.