[CogitoCogito]

> Rationals are conintuous, but not a continuum.

No. In fact, your claim is contradicted by your own link. As stated by Asaf Karagila in the comments:

> Continuity is a property of functions. You seem to ask why the rational numbers are not connected (or path connected).

But you are correct that you don't need real numbers to describe continuous functions. As you link points out, one is a property of spaces (or domains or whatever you want to call it) and the other is a property of function.

However, talking about continuous functions between complete spaces (basically "complete" is what makes the real numbers "real") is extremely natural and basically goes hand in hand with continuous functions. It really ties together a lot of the theory if you're talking about metric continuity.

Regardless, you also don't _lose_ anything by talking about real numbers. You can of course use it as a tool to develop a theory and then choose to apply the results to the rational numbers (or algebraic, etc.).