Now I see this question as naturally applying Riemann's condition, which states that $f$ is Riemann integrable iff $U(f,P) - L(f,P) < \varepsilon$. I think this kind of automatically solves itself in the forward direction if I am not mistaken.The key question being, can it be any partition? (as in $S$ and $P$ both work, and so can be plugged in at will?)
Conversely, same logic applied in reverse means it's Riemann integrable. I am just unsure, I feel I am skipping steps.