In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; though some authors do not distinguish between set-valued functions and multifunctions. A multivalued function of sets f : X → Y is a subset
Γ f ⊆ X × Y . {\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.}
Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf. If f is an ordinary function, it is a multivalued function by taking its graph
Γ f = { ( x , f ( x ) ) : x ∈ X } . {\displaystyle \Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.}
They are called single-valued functions to distinguish them.
Motivation
The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f ( z ) {\displaystyle f(z)} in some neighbourhood of a point z = a {\displaystyle z=a} . This is the case for functions defined by the implicit function theorem or by a Taylor series around z = a {\displaystyle z=a} . In such a situation, one may extend the domain of the single-valued function f ( z ) {\displaystyle f(z)} along curves in the complex plane starting at a {\displaystyle a} . In doing so, one finds that the value of the extended function at a point z = b {\displaystyle z=b} depends on the chosen curve from a {\displaystyle a} to b {\displaystyle b} ; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function. For example, let f ( z ) = z {\displaystyle f(z)={\sqrt {z}}\,} be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of z = 1 {\displaystyle z=1} in the complex plane, and then further along curves starting at z = 1 {\displaystyle z=1} , so that the values along a given curve vary continuously from 1 = 1 {\displaystyle {\sqrt {1}}=1} . Extending to negative real numbers, one gets two opposite values for the square root—for example ±i for −1—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 nth roots, logarithms, and inverse trigonometric functions. To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the principal value, 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 (monodromy). These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function f ( z ) {\displaystyle f(z)} as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to f ( z ) {\displaystyle f(z)} .
Inverses of functions If f : X → Y is an ordinary function, then its inverse is the multivalued function
Γ f − 1 ⊆ Y × X {\displaystyle \Gamma _{f^{-1}}\ \subseteq \ Y\times X}
defined as Γf, viewed as a subset of X × Y. When f is a differentiable function between manifolds, the inverse function theorem gives conditions for this to be single-valued locally in X. For example, the complex logarithm log(z) is the multivalued inverse of the exponential function ez : C → C×, with graph
Γ log ( z ) = { ( z , w ) : w = log ( z ) } ⊆ C × C × . {\displaystyle \Gamma _{\log(z)}\ =\ \{(z,w)\ :\ w=\log(z)\}\ \subseteq \ \mathbf {C} \times \mathbf {C} ^{\times }.}
It is not single valued, given a single w with w = log(z), we have
log ( z ) = w + 2 π i Z . {\displaystyle \log(z)\ =\ w\ +\ 2\pi i\mathbf {Z} .}
Given any holomorphic function on an open subset of the complex plane C, its analytic continuation is always a multivalued function.
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