Definition
Given an arbitrary set
, a totally ordered set
, and a function,
, the
over some subset
of
is defined by

If
or
is clear from the context, then
is often left out, as in
In other words,
is the set of points
for which
attains the function's largest value (if it exists).
may be the empty set, a singleton, or contain multiple elements.
In the fields of convex analysis and variational analysis, a slightly different definition is used in the special case where
are the extended real numbers. In this case, if
is identically equal to
on
then
(that is,
) and otherwise
is defined as above, where in this case
can also be written as:

where it is emphasized that this equality involving
holds only when
is not identically
on
.
Arg min
The notion of
(or
), which stands for argument of the minimum, is defined analogously. For instance,

are points
for which
attains its smallest value. It is the complementary operator of
.
In the special case where
are the extended real numbers, if
is identically equal to
on
then
(that is,
) and otherwise
is defined as above and moreover, in this case (of
not identically equal to
) it also satisfies:

Examples and properties
For example, if
is
then
attains its maximum value of
only at the point
Thus

The
operator is different from the
operator. The
operator, when given the same function, returns the maximum value of the function instead of the point or points that cause that function to reach that value; in other words
is the element in 
Like
max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike 
may not contain multiple elements: [note 2] for example, if
is
then
but
because the function attains the same value at every element of 
Equivalently, if
is the maximum of
then the
is the level set of the maximum:

We can rearrange to give the simple identity [note 3]

If the maximum is reached at a single point then this point is often referred to as the
and
is considered a point, not a set of points. So, for example,

(rather than the singleton set
), since the maximum value of
is
which occurs for
[note 4] However, in case the maximum is reached at many points,
needs to be considered a set of points.
For example

because the maximum value of
is
which occurs on this interval for
or
On the whole real line
so an infinite set.
Functions need not in general attain a maximum value, and hence the
is sometimes the empty set; for example,
since
is unbounded on the real line. As another example,
although
is bounded by
However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty 