How could an element of a real segment $]-r,r[$ NOT be a limit point of the segment?

Question

Rudin's Principles of mathematical analysis states

Doesn't the fact that $S$ be a segment of $\mathbb{R}$ by definition imply that every point of $S$ is a limit point of $S$? If this is the case, how could there be an element $x\in S$ such that $x$ is not a limit point of $S$?