Taylor's Theorem for functions on the Rational Numbers

I have been looking for different proofs of the Taylor's Theorem with Peano form of the remainder. But all proofs I have found use some form of the Mean Value Theorem or L'Hôpital's Rule (which also seems to need some form of the Mean Value Theorem).

Now, the Mean Value Theorem depends on the topology of $\mathbb{R}$. I used to believe that the Peano form of the remainder could be proved by more elementary means, only using the definition of the derivative (which is equivalent to Taylor's Theorem for $n = 1$).

So I ask: is the theorem true for a function $f\colon \mathbb{Q} \to \mathbb{Q}$? Or does it depend on some property of $\mathbb{R}$?