Extreme Value Theorem

The extreme value theorem states that 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.