Characterising continuity of a Riemann integrable function in terms of the upper and lower sums

Question

Given a bounded Riemann integrable function $f:[a,b] \to \mathbb{R}$, I would like to know if there is any way to characterise the continuity of $f$ in terms of the difference between the upper Riemann sum $U(f,P_n)$ and the lower Riemann sum $L(f,P_n)$.