Order-Convex Subset of the Real Line

Let $\mathbb{R}$ be the set of real numbers, with the order $\le$ of its ordered field structure.

A subset $J\subseteq\mathbb{R}$ is order-convex if for all $x,z\in J$ and every $y\in\mathbb{R}$ with $x\le y$ and $y\le z$, one has

$$y\in J .$$