Since f is piecewise continuous, we can find a parititon P such that f∣J is continuous on J for ∀J∈P. By Proposition 11.5.3, we have f∣J is Riemann integrable on J. We define FJ(x)={f∣J(x),0,x∈Jx∈I\J By Theorem 11.4.1(g), FJ is Riemann integrable on I, and we further have f(x)=J∈P∑FJ(x) So by Theorem 11.4.1(a), f is Riemann integrable on I.