If $\underline{ab}$ is a region in $C$, then: $ C = \{ x \mid x < a \} \cup \{a\} \cup \underline{ab} \cup \{b \} \cup \{ x \mid b < x \}. $
Where C is a continuum that is nonempty, has no first or last point, and is ordered $<$.
Regions can be defined as all the points between $a$ and $b$ (such that $a) denoted by $\underline{ab}$.
This seems a bit obvious to me, but perhaps the proof is more clear. I thought of trying to prove each possible point would end up being some point on $C$ and that $\underline{ab}$ is also a continuum, but I'm not sure this is the way to go.