Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 6 / 1

A continuous finite-variation path has zero quadratic variation. Hence a continuous finite-variation martingale $M$ satisfies $⟨M⟩=0$. After localizing to make it square-integrable,Letting the localization level tend to infinity shows that $M_t=M_0$ for every $t$ almost surely; continuity makes the equality simultaneous in $t$.