A multi-valued function$f: A \to B$ is like a function from $A$ to $B$ except that there may be more than one possible value $f(x)$ for a given element $x$ of $A$. (Compare a partial function, where $f(x)$ may not exist at all.)

In older literature (into the 20th century, especially in analysis), functions were often considered to be multi-valued by default, requiring one to specify a singe-valued function otherwise. As set-theoretic formalisation spread, this intuition became difficult to maintain, and the modern concept of function must be single-valued. If you want multi-valued functions, then you can get them in terms of single-valued functions as below.

Definitions

Given sets $A$ and $B$, a multi-valued function$f$ from a $A$ to $B$ is a span

$\array {
& & D \\
& \swarrow_\pi & & \searrow^f \\
A & & & & B \\
}$

of single-valued functions, where $\pi: D \to A$ is a surjection. (This condition can be dropped to define a multi-valued partial function, which is simply a span.)

We will call $A$ and $B$ the source and target of $f$ as usual; then we call $D$ the domain of $f$ and $\pi: D \to A$ the projection of the domain onto the source. By abuse of notation, the multi-valued function $f$ is conflated with the (single-valued) function $f: D \to B$.

Often one can assume that the induced function $D \to A \times B$ is an injection; in that case, a multi-valued function is the same as an entire relation. On the other hand, if you're considering all of the multi-valued functions for a given $D$, then this restriction is not really appropriate.

We consider two multi-valued functions (with the same given source and target) to be equal if there is a bijection between their domains that makes the obvious diagrams commute.

Examples

In 19th-century analysis, one considered the square-root function, the logarithm, and so forth to be multi-valued functions of complex numbers. We now understand this in terms of Riemann surfaces; the domain $D$ above is a Riemann surface. (Notice that the logarithm is actually a multi-valued partial function from $\mathbf{C}$ to $\mathbf{C}$, although it is a multi-valued total function on $\mathbf{C} \setminus \{0\}$.)