I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real numbers, the Cauchy completeness (or, equivalently, the existence of the least upper bound). But still, it appears you could get quite far in $\Bbb Q$. First of all, elementary notions like continuity, limits and differentiability are easily definable (though even among the real functions that can be restricted to operators on $\Bbb Q$, not all of them will satisfy these definitions). Riemann integrability can also be defined in a limited form: In real analysis, the definition of the integral involves the supremum and the infimum. But we can define rational Riemann integrability if the upper and lower bounds in question do exist in $\Bbb Q$ and satisfy the appropriate relations, which will of course often be the case. But obviously, a function like $\Bbb Q\setminus\{0\}\ni x\mapsto 1/x$ is not integrable, since $\int_a^b \tfrac 1 x d x = \log(\tfrac b a)$ is not rational for unequal $a,b\in\Bbb Q$. But still, it seems we can get quite far.
Does someone have an impression of how far we can get? Also, I have been totally unable to find any mention of rational analysis, as poor as it may be, in literature. Seeing that how far I could get, I find that strange. Does someone have an opinion about why this is the case.
