Editing Function (mathematics) (section)

== Multi-valued functions ==
{{main|Multi-valued function}}
[[File:Function with two values 1.svg|thumb|right|Together, the two square roots of all nonnegative real numbers form a single smooth curve.]]
[[File:Xto3minus3x.svg|thumb|right]]
Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a [[neighbourhood (mathematics)|neighbourhood]] of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point <math>x_0,</math> there are several possible starting values for the function.

For example, in defining the [[square root]] as the inverse function of the square function, for any positive real number <math>x_0,</math> there are two choices for the value of the square root, one of which is positive and denoted <math>\sqrt {x_0},</math> and another which is negative and denoted <math>-\sqrt {x_0}.</math> These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single [[smooth curve]]. It is therefore often useful to consider these two square root functions as a single function that has two values for positive {{mvar|x}}, one value for 0 and no value for negative {{mvar|x}}.

In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the [[implicit function]] that maps {{mvar|y}} to a [[root of a function|root]] {{mvar|x}} of <math>x^3-3x-y =0</math> (see the figure on the right). For {{math|1=''y'' = 0}} one may choose either <math>0, \sqrt 3,\text{ or } -\sqrt 3</math> for {{mvar|x}}. By the [[implicit function theorem]], each choice defines a function; for the first one, the (maximal) domain is the interval {{closed-closed|−2, 2}} and the image is {{closed-closed|−1, 1}}; for the second one, the domain is {{closed-open|−2, ∞}} and the image is {{closed-open|1, ∞}}; for the last one, the domain is {{open-closed|−∞, 2}} and the image is {{open-closed|−∞, −1}}. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single ''multi-valued function'' of {{mvar|y}} that has three values for {{math|−2 < ''y'' < 2}}, and only one value for {{math|''y'' ≤ −2}} and {{math|''y'' ≥ −2}}.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically [[analytic function]]s. The domain to which a complex function may be extended by [[analytic continuation]] generally consists of almost the whole [[complex plane]]. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets {{mvar|i}} for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets {{math|−''i''}}. There are generally two ways of solving the problem. One may define a function that is not [[continuous function|continuous]] along some curve, called a [[branch cut]]. Such a function is called the [[principal value]] of the function. The other way is to consider that one has a ''multi-valued function'', which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the [[monodromy]].