In mathematics, multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function.

Multivalued functions arise commonly in applications of implicit function theorem, since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the inverse function of a differentiable function is a multivalued function. For example, the complex logarithm is a multivalued function, as the inverse of the exponential function. It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at 0, one gets another value than the starting one after a complete turn. This phenomenon is called monodromy.

Another common way for defining a multivalued function is analytic continuation, which generates commonly some monodromy: analytic continuation along a closed curve may generate a final value that differs from the starting value.

Multivalued functions arise also as solutions of differential equations, where the different values are parametrized by initial conditions.