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I was watching this video from Michael Penn where he was showing how standard analysis 1 results fail if we restrict functions to $\mathbb{Q}$.

After I've been thinking about it for a bit, it seems to me that basic results can be rescued with a few modifications, since a real number can be thought of a (equivalence class of) cauchy sequence of rational numbers.

Continuity can be defined as follows: f is continuous if, for every cauchy sequence $x_n, f(x_n)$ is a cauchy sequence as well.

The IVT can be restated as follows: Let f be continous, and assume there are $a \neq b$ s.t. $f(a)f(b) < 0$, then there's a cauchy sequence $c_n$ with $a < c_n < b \ \forall n,$ and $f(c_n) \to 0$.

One potentially important difference I see is that now continuity is not a local property anymore.

Given the above, I was curious if the construction actually fails in some important way, or if the real numbers are, indeed, not indispensable, for doing analysis.

Disclaimer: I'm not claiming that real numbers are useless or should not be used. I believe standard analysis is simpler, and much more intuitive and elegant than "rational" analysis.

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Mathematics Meta, or in Mathematics Chat. Comments continuing discussion may be removed. $\endgroup$ Commented Apr 26 at 21:50

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Continuity can be defined as follows...

In order for your theory to coexist with real analysis and topology, you probably don't want to overload the word "continuous" to describe these functions on the rational numbers. It would be better to say "Cauchy-continuous" or "Cauchy-regular". These are attested on Wikipedia, at least: https://en.wikipedia.org/wiki/Cauchy-continuous_function

Of course, writing "Cauchy" everywhere does kind of give the game away.

[functions like sin and exp] can be defined through power series, just as in classical analysis

Those power series don't converge in $\mathbb Q$ though. They will instead give you functions $\mathbb Q \to \mathbb R$, so you end up using the real numbers.

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  • $\begingroup$ I didn't consider how this might impact the other areas of math. That's a good point. Thanks! $\endgroup$ Commented Apr 26 at 18:01

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