 Solution

Let $C$ be a component of the open set $X\subseteq \mathbb{R}$. Being a connected subset of the real line, $C$ is an interval. For any $x\in C$, openness of $X$ gives $ϵ>0$ with $(x-ϵ,x+ϵ)\subseteq X$. This interval is connected and meets $C$, so their union is connected. Maximality of $C$ makes the entire interval lie in $C$. Thus every point of $C$ is interior, so each component is an open interval, allowing unbounded endpoints.