Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 1 / c / Solution

For any partition of $[0,t]$,The final sum is at most the largest increment of $f$ times its total variation. It tends to zero because $f$ is uniformly continuous. Part b then gives the integration-by-parts identityThus the formula stated in the question holds when $f(0)=0$; for a general initial value the necessary endpoint correction is $-f(0{)}^2$.