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Concept
Extreme value theorem
Summary
In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed interval , then must attain a maximum and a minimum, each at least once. That is, there exist numbers and in such that:
The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function on the closed interval is bounded on that interval; that is, there exist real numbers and such that:
This does not say that and are necessarily the maximum and minimum values of on the interval which is what the extreme value theorem stipulates must also be the case.
The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value. Both proofs involved what is known today as the Bolzano–Weierstrass theorem.
The following examples show why the function domain must be closed and bounded in order for the theorem to apply. Each fails to attain a maximum on the given interval.
defined over is not bounded from above.
defined over is bounded but does not attain its least upper bound .
defined over is not bounded from above.
defined over is bounded but never attains its least upper bound .
Defining in the last two examples shows that both theorems require continuity on .
When moving from the real line to metric spaces and general topological spaces, the appropriate generalization of a closed bounded interval is a compact set. A set is said to be compact if it has the following property: from every collection of open sets such that , a finite subcollection can be chosen such that .
Official source
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