Realvalued function/Extrema/Introduction/Section

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Definition  

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.


Definition  

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

holds.

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