TITLE: Why are the rational numbers not continuous?
QUESTION [18 upvotes]: I am trying to understand why the rational numbers are not continuous.
Given two rational numbers $a$ and $b$, I can always find a number $c = \frac{a+b}{2}$ between these two numbers. So when I plot the rational numbers as a line, this is a steady line (unlike natural numbers, which are obviously not continuous). Why are they not continuous?

REPLY [4 votes]: Maybe I can help to tidy up some of the terminology. A linearly ordered set is called a linear continuum if it has no gaps or jumps. A jump is a cut where both sides have endpoints; an ordering without jumps is called dense. A gap is a cut where neither side has an endpoint; an ordering without gaps is called complete.
The natural numbers have jumps, but no gaps; they are complete, but not dense. The rational numbers have gaps, but no jumps; they are dense, but not complete. The real numbers have neither gaps nor jumps; they are a continuum.
Some authors (and translators from other languages) use the English adjective "continuous" to describe a continuum. As some of the comments and other answers point out, this usage may be confusing, since the adjective "continuous" is also used for functions, and the meaning isn't quite the same.