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What kind of algebraist "refuses to acknowledge the existence of any characteristic 0 field other than $\mathbb{Q}$"?? But there is a good question in here nevertheless: the basic definitions of limit, continuity, and differentiability all make sense for functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$. The real numbers are in many ways a much more complicated structure than $\mathbb{Q}$ (and in many other ways are much simpler, but never mind that here!), so it is natural to ask whether they are really necessary for calculus.

Strangely, this question has gotten serious attention only relatively recently. For instance:

$\bullet$ Tom Korner's real analysis text takes this question seriously and gives several examples of pathological behavior over $\mathbb{Q}$.

$\bullet$ Michael Schramm's real analysis text is unusually thorough and lucid in making logical connections between the main theorems of calculus (though there is one mistaken implication there). I found it to be a very appealing text because of this.

$\bullet$ My honors calculus notes often explain what goes wrong if you use an ordered field other than $\mathbb{R}$.

$\bullet$ $\mathbb{R}$ is the unique ordered field in which real induction is possible.

$\bullet$ The most comprehensive answers to your question can be found in two recent Monthly articles, by Jim Propp and by Holger Teismann.

But as the title of Teismann's article suggests, even the latter two articles do not complete the story.

$\bullet$ Here is a short note whose genesis was on this site which explains a further pathology of $\mathbb{Q}$: there are absolutely convergent series which are not convergent.

$\bullet$ Only a few weeks ago Jim Propp wrote to tell me that Tarski's Fixed Point Theorem characterizes completeness in ordered fields and admits a nice proof using Real Induction. (I put it in my honors calculus notes.) So the fun continues...