$f$ is integrable on $[a,b] \iff \forall\epsilon>0, \exists P$ of $[a,b]$ such that
$$U(f,P) - L(f,P) < \epsilon$$
I was thinking that a better definition would be if $U(f,P) = L(f,P)$, but I was corrected that it wouldn't work well for a curve like $f(x) = x$. The geometry just doesn't work.
On the other bound, saying that given any positive number, I can find a partition such that the difference $U(f,P) - L(f,P) < \epsilon$ is bounded doesn't feel like a strong enough condition for integrability. Isn't the goal of analysis is always to make $\epsilon$ as small as possible, and possibly $0$?
Please see my other question on A terminology to analysts for possible relevance. Thank you