We want a set which corresponds to our notion of points on a line.1
We call a subset $R$ of $\Q $ a rational cut if (a) $R \neq \varnothing$, (b) $R \neq \Q $, (c) for all $q \in R$, $r \leq q \implies r \in R$, and (d) $R$ has no greatest element. Briefly, the intuition is that the point is the set of all rationals to less than (or, potentially, equal to) some particular rational number.2
We follow tradition and denote the set of real numbers by $\R $, likely a mnemonic for “real.”
Some authors call a real number a quantity or a continuous quantity. The real numbers, then, are said to be continuous. When contrasting (using this terminology) a finite set with the real numbers, one refers to the finite set as discrete.3