ABSTRACT
In this chapter, the author aims to prove a theorem: the Intermediate Value Theorem. This theorem relies on two major assumptions: the continuity of the function, and a topological property of the interval. The chapter provides a brief discussion of what it means for a subset of R to be connected. It turns out that the most efficient way to describe the intuitive concept of “connectedness” is to first define disconnectedness. The definition of “connectedness” is unwieldy because it can be difficult to leverage or to show directly that a set cannot be split into a union of two nonempty separated sets. Because of this, proofs involving connectedness are often structured as proofs of the contrapositive or proofs by contradiction.
