ABSTRACT
This chapter proves (arguably) one of the most important theorems in Real Analysis: the Mean Value Theorem. To get there, one needs another result first: the Extreme Value Theorem, which is very important in its own right. There are a number of strategies one can employ to introduce and prove the Extreme Value Theorem and the Mean Value Theorem. To prove these theorems most efficiently, a new concept that is closely related to the idea of compactness is introduced. The first major theorem shows that continuous functions preserve the property of compactness, in the sense that the image of a compact set (under a continuous function) remains compact. There is a “topological” approach to prove that the continuous image of a compact set is compact. When it comes to the interplay between pre-images and unions, one can prove a stronger statement involving the union of an arbitrary collection of (finitely or infinitely many) sets.
