A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from 
to 
is an object 
such that every 
is uniquely associated with
an object 
.
A function is therefore a many-to-one (or sometimes
one-to-one) relation.
The set 
of values at which a function is defined is called its domain, while the set 
of values that the function can produce is called
its range. Here, the set 
is called the codomain of 
.
In the context of univariate, real-valued functions 
,
the fact that domain elements
are mapped to unique range elements can be expressed graphically by way of the vertical
line test.
In some literature, the term "map" is synonymous with function. Some caution must be exhibited, however, as it is not uncommon for the term map to denote a function with some sort of unspoken regularity assumption, e.g., in point-set topology, where "map" sometimes refers to a function which is continuous with respect to some topology.
Examples of functions over the real numbers 
include the sine 
(many-to-one), 
(one-to-one), 
(two-to-one except for the single point 
), etc.
Unfortunately, the term "function" is also used to refer to relations that map single points in the domain to possibly multiple points in the range. These "functions" are called multivalued functions (or multiple-valued functions), and arise prominently in the theory of complex functions, where the presence of multiple values engenders the use of so-called branch cuts.
Several function notations are commonly used to represent (non-multivalued) functions. The notation 
specifies the domain 
and codomain 
, while 
specifies the value assigned
to an element 
. The argument 
need not be a number; it may, for
example, be a vector or an ordered list.
For example, "
,
where 
" specifies a univariate
function on the real numbers.
Generally speaking, the symbol 
refers to the function itself, while 
refers to the value taken
by the function when evaluated at a point 
. However, especially in more introductory texts, the notation 
is commonly used to refer to the
function 
itself (as opposed to the value of the function evaluated
at 
). In this context, the argument 
is considered to be a dummy
variable whose presence indicates that the function 
takes a single argument (as opposed
to 
, etc.). While this notation
is deprecated by professional mathematicians, it is the more familiar one for most
nonprofessionals. Therefore, unless indicated otherwise by context, the notation 
is taken in this work to be a shorthand
for the more rigorous 
.
