This video explores the possibility of defining calculus over the rational numbers.
He glosses over the definitions of "continuous". For example 13 minutes in he has
$f(x)=x^2-2$
Note: $f$ is continuous on $[0,2]\cap\mathbb{Q}$
In fact, he opens with a definition
Recall: a function $f$ is continuous at a if $\lim_{x\to a}f(x)=f(a)$
and then immediately gives an example of how this doesn't work on rationals. But then goes on to use "continuous" in his exploration anyway.
What does "continuous" mean in the context of rational numbers? Is he just exploring nonsense statements?
The $\epsilon$-$\delta$ definition of limit, and of continuity, make sense in the rationals. Many (but not all) of the proofs you may see about limits and continuity in your calculus course, make sense in any ordered field.
More generally (as you may learn in the future) they make sense in any "metric space".