TITLE: Study of functions from $\mathbb{Q}$ to $\mathbb{Q}$
QUESTION [6 upvotes]: Is it possible to study functions from $\mathbb{Q}$ to $\mathbb{Q}$ with ordinary calculus ?
Obviously with the limitation that $\mathbb{Q}$ is not complete.
So much less limits, derivatives and integrals exist; but does it make sense a tangent in $\mathbb{Q^2}$

REPLY [3 votes]: It could make some sense, but it will be quite pathological. $\bf Q$ is not a differentiable manifold, so differentiation in the usual sense doesn't make much sense, furthermore, there is no nontrivial continuous measure on $\bf Q$ (because $\bf Q$ is countable), so Lebesgue integral will be pretty much completely useless in studying it.
If course, you can still calculate limits, derivatives and Riemann integrals as if you were working in $\bf R$. After all, limits make sense in any topological space, including $\bf Q$, and all these objects are defined as limits, so at worst, they may just fail to exist, which can lead to rather pathological examples.
For instance, the function $1/x$ will still be continuous except in $0$, but it will be very non-integrable (as the logarithm of a positive rational number distinct from $1$ is irrational). It is not hard to imagine a function $f:{\bf Q}\to{\bf Q}$ and three rational numbers $a<b<c$ such that the “integral” from $a$ to $c$ is well-defined, but not from $a$ to $b$ (because the integral from $a$ to $b$ converges to an irrational number, but from $a$ to $c$ to a rational one).

REPLY [2 votes]: On the other hand, in $\mathbb Q$ (and, indeed, in other fields) use of the formal derivative for a polynomial is useful for purely algebraic purposes.  Establishing when a polynomial has multiple roots, for example.