I don't see how the inf of the function on that interval would be 0 since it can take on negative values on such an interval. But I guess the general idea would be that \(\displaystyle U(f, [0,1]) = U(f,[0,a]) + U(f,[a,1])\) and the same for the lower sums, correct? (I know that is technically an incorrect statement since we have to state that the union of the two partitions is in fact the partition of the larger interval and that a is an element in both partitions, etc, etc).