 Solution

Given $\epsilon >0$, choose a partition of $[a,d]$ whose upper-minus-lower sum is below $\epsilon$, and refine it by inserting $b,c$. The contribution from subintervals lying in $[b,c]$ is nonnegative and no larger than the total difference. Restricting the refined partition to $[b,c]$ therefore gives upper-minus-lower sum below $\epsilon$, so the criterion proves integrability there.