There are many ways to define the Riemann Integral. I am using this one, where I denote $\sigma(f,P^{*})$ the Riemann Sum relative to a tagged partition $P^{*}$:
$\textbf{Definition}$
We say that a function $f:[a,b] \to \mathbb{R}$ is Riemann-Integrable if exist the limit:
$I=\lim_{||P|| \to 0} \sigma(f,P^{*})$
and then we write $I=\int_{a}^{b}f(x)dx$.
The limit exist in the sense that given $\epsilon > 0$, there is $\delta > 0$ such that for any partition $P$ of $[a,b]$ with $||P|| < \delta$ and for any tagged partition $P^{*}$, we have:
$$|\sigma(f,P^{*}) - I| < \epsilon$$
By definition, if $f$ is integrable in $[a.b]$, then given $\epsilon < 0 $ exists two tagged partition $P^{*}$ and $P^{**}$ such that:
$$|\sigma(f,P^{*}) - I| < \epsilon / 2$$
$$|\sigma(f,P^{**}) - I| < \epsilon / 2$$
hence, $|\sigma(f,P^{*}) - \sigma(f,P^{**})| < \epsilon$. Therefore, is a necessary condition for integrability that one can find a partition $P$ such that two Riemann Sums relative to $P$ are very close together, no matter what the scalars that we pick in $P^{*}$ and $P^{**}$.
I am pretty sure that this is a sufficient condition as well. But I have any ideas how to prove it. Can anyone help with this?