A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being one-to-one or many-to-one, the usage is so common that there is no way to dislodge it. When considering multivalued functions, it is therefore necessary to refer to usual "functions" as single-valued functions.
While the trigonometric, hyperbolic, exponential, and integer power
functions are all single-valued functions,
their inverses are multivalued. For example, the function 
maps each complex number 
to a well-defined number 
, while its inverse function 
maps, for example, the value 
to 
. While a unique principal
value can be chosen for such functions (in this case, the principal
square root is the positive one), the choices cannot be made continuous
over the whole complex plane. Instead, lines of
discontinuity must occur.
The discontinuities of multivalued functions in the complex plane are commonly handled through the adoption of branch cuts, but use of Riemann surfaces is another possibility.
