Weierstrauss Theorem

Definition: The Weierstrauss Theorem is that a continuous function on a closed and bounded set will have a maximum and a minimum.

This theorem is often used implicitly, in the assumption that some set is compact, meaning closed and bounded. Examples that may help clarify:

Example 1: Consider a set which is unbounded, like the real line. Say variable x has any value on the real line, and we wish to maximize the function f(x)=2x. It doesn't have a maximum or minimum because values of x further from zero have more and more extreme values of f(x).

Example 2: Consider a set which is not closed, like (0,1). Again, let f(x) be 2x. Again this function has no maximum or minimum because there is no largest or smallest value of x in the set.

(Econterms)

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