**Definition: **A (Real) single or multivalued function, $f$ , of a single or many real
variables is a correspondence between a set of controlling
real numbers $x_1, x_2, ..., x_n$ (called the domain of the function) and related
controlled real number values $y_1,y_2,...,y_k$ that are determined uniquely
by the controlling numbers $x_1, x_2, ..., x_n$.

Each $n$- tuple, $(x_1, x_2, ..., x_n)$ in the domain of the function corresponds to one and only
one $k$ - tuple, $(y_1,y_2,...,y_k)$ determined by the function, denoted:
$ f (x_1, x_2, ..., x_n) = (y_1,y_2,...,y_k)$.

This is sometimes denoted by writing $f: x \rightarrow y =
f(x)$ where $x = (x_1, x_2, ..., x_n)$ and $y = (y_1,y_2,...,y_k)$.