Left-total

In mathematics, a multivalued function from a domain X to a codomain Y is, roughly speaking, a function which may associate more than one element of Y to each element of X. Formally, a multivalued function is an ordinary function from X to the set of the nonempty subsets of Y, or a left-total relation.

In this context, an ordinary function is often called a single-valued function to avoid confusion.

The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function ${\displaystyle f(z)}$ in some neighbourhood of a point ${\displaystyle z=a}$. This is the case for functions defined by the implicit function theorem or by a Taylor series around ${\displaystyle z=a}$. In such a situation, one may extend the domain of the single-valued function ${\displaystyle f(z)}$ along curves in the complex plane starting at ${\displaystyle a}$. In doing so, one finds that the value of the extended function at a point ${\displaystyle z=b}$ depends on the chosen curve from ${\displaystyle a}$ to ${\displaystyle b}$; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function. For example, let ${\displaystyle f(z)={\sqrt {z}}\,}$ be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of ${\displaystyle z=1}$ in the complex plane, and then further along curves starting at ${\displaystyle z=1}$, so that the values along a given curve vary continuously from ${\displaystyle {\sqrt {1}}=1}$. Extending to negative real numbers, one gets two opposite values of the square root such as ${\displaystyle {\sqrt {-1}}=\pm i}$, 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.

...
Wikipedia