From

Revuz and Yor - Continuous Martingales and Brownian Motion 1999

Chapter IV Proposition 1.13

it is proven, that for a continuous local martingale $M_t$ the intervals of constancy are equal with those of the predictable quadratic variation $<M>_t$ or optional quadratic variation $[M]_{t}$ (since they coincide due to the continuity of the local martingale).

I wonder if this stays true for $M_{t}$ being just càdlàg. I guess no.

So lets consider this setup: Given a square integrable Martingale $X_t=F_t-a\cdot K_{t}$ with predictable quadratic variation $b\cdot K_{t}$ where $a,b$ are constants and $K_{t}$ is continuous but $F_{t}$ only càdlàg. With the aim to conclude from $K_{t}$ being constant on some interval (predictable quadratic variation is continuous process of $K_{t}$ being constant) implies that $X_{t}$ is constant on that interval and thus $F_{t}$ on the interval. Where $K_{0}=0$, $K_{t}\rightarrow \infty$ a.s. and a non decreasing process.