From Kal97, pg. 446:
Theorem 23.14 Any semimartingale $X$ has an a.s. unique decomposition $X=X_0 + X^c + X^d$ where $X^c$ is a continuous local martingale with $X_0^c=0$ and $X^d$ is a purely discontinuous semimartingale.
Q: Is it true that if $X$ is a local martingale then both $X^c$ and $X^d$ are local martingales? If not, does anyone have a counterexample?
My intuition tells me it is true. But I have a very weak handle on this material and cannot find any theorems/corollaries/remarks/etc. in Kal97 or Pro05 that definitively say whether this is the case or not.