Topology: An Inquiry-Based Approach

In this section we present two important consequences of connectedness. The first consequence is the Intermediate Value Theorem. In calculus, the Intermediate Value Theorem tells us that if $f$ is a continuous function on a closed interval $[a,b]\text{,}$ then $f$ assumes all values between $f(a)$ and $f(b)\text{.}$ This result seems straightforward if one just draws a graph of a continuous function on a closed interval. But a picture is not a proof. We state and then prove a more general version of the Intermediate Value Theorem.

Our second consequence of connectedness involves a fixed point theorem. Fixed point theorems are important in mathematics. A fixed point of a function $f$ is an input $c$ so that $f(c)=c\text{.}$ There are many fixed point theorems — one of the most well-known is the Brouwer Fixed Point Theorem that shows that every continuous function from a closed ball $B$ in ${\mathbb{R}}^n$ to itself must have a fixed point. We can use the Intermediate Value Theorem to prove this result in $\mathbb{R}\text{.}$