>Rational numbers are continuous, by the typical definition. Between
>any two rational numbers there is another (and therefore, an infinite
>number of others).
This is density. Q is dense indeed, but highly discontinuous.
Continuity means either that all dedekind-cut define numbers, or that
all cauchy sequences define numbers.
Note that in classical analysis these two definitions are equivalent,
but in intuitionistic mathematics they are not!
Unlike computability, continuity like provability is a relative