In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea of connectedness?
High wow-ratio examples are specially welcomed… 🙂
Best Answer
If $h:[a,b]\to R$ is continuous and one-to-one, then $h$ is monotone.
Proof: The image of the connected set $\{(s,t): a \le s < t \le b\}$ under the map $h(t)-h(s)$ is a connected subset of $R\setminus\{0\}$.