I'm going through *Problem and Theorems in Classical Set Theory* by Komjath and Totik, and the answer to this question is:
Let $[x]$ (resp. $\{x\}$) denote the integral (resp. fractional) part of $x$, and set $x ≺ y$ if $\{x\} < \{y\}$ or if $\{x\} = \{y\}$ and $[x] < [y]$.

It is clear that in this ordering $x − 1$ is the predecessor, and $x + 1$ is the successor of $x$.

Everything I had read states that elements of the the reals $\mathbb{R}$ don't have an immediate successors or predecessor. Could someone help explain this? Thanks.