Extreme Value Theorem

If a function  is continuous on a closed interval , then  has both a maximum and a minimum on . If  has an extremum on an open interval , then the extremum occurs at a critical point. This theorem is sometimes also called the Weierstrass extreme value theorem.

The standard proof of the first proceeds by noting that  is the continuous image of a compact set on the interval , so it must itself be compact. Since  is compact, it follows that the image  must also be compact.