A multifunction is clearly not a function in the usual sense, as functions may only take a single value at a given argument. However, such multifunctions arise naturally, particularly in complex analysis, and various approaches can be taken to describe them.

Common examples are square roots, inverse trigonometric functions, and the complex logarithm. For a positive real x, there are two choices of a square root $\pm \sqrt{x}$, and the function $\sqrt{x}$ is well defined, as a choice is made to take the positive square root as the principal value. Likewise, cos⁻¹x is well defined on [–1,1] by taking principal values in [0,π‎].

For the complex square root, aside from the positive real axis, there are no obvious principal values. Further, it is impossible to continuously (see continuous function) extend the notion of $\sqrt{x}$ for the positive reals to the entire complex plane. One approach is to limit the domain to a cut plane and define a branch (a continuous set of principal values) on the cut plane, which may lead to discontinuities across the cut.

Alternatively, all the possible values of a multifunction may be considered simultaneously on a Riemann surface. So the Riemann surface of $\sqrt{z}$ is the surface w² = z in ℂ². On such a surface will be present both the points (z, $\sqrt{z}$) and (z, $-\sqrt{z}$).