# Realvalued function/Extrema/Introduction/Section

Let denote a set, and

a
function.
We say that attains in a point
its *maximum*, if

and that attains in its *minimum*, if

The common name for a maximum or a minimum is *extremum*. In the preceding definition, we also talk about the *global maximum*, since the property refers to all elements of the domain of the definition. When we are only interested in the behavior on an open, maybe small, neighborhood, then the concept of a local maximum is relevant.

Suppose that is a subset and let

denote a
function.
We say that attains a *local maximum* in a point
,
if there exists some
,
such that for all
fulfilling
,
the estimate

holds. We say that attains a *local minimum* in a point
,
if there exists some
,
such that for all
fulfilling
,
the estimate

If
holds for all
,
then we talk about an *isolated maximum*.