= Solution
A \b[<continuous martingale>] is an <adapted process> with almost surely continuous paths, $\mathbb E|M_t|<\infty$ for every $t$, and
$$
\boxed{\mathbb E[M_t\mid\mathcal F_s]=M_s\qquad(s\leq t).}
$$
A \b[<continuous local martingale>] is continuous and <adapted> and has a <localizing sequence> of <stopping times> $T_n\uparrow\infty$ almost surely such that every $M^{T_n}_t=M_{t\wedge T_n}$ is a true <martingale>. The initial value is normally integrable; the frequently used class $\mathcal M_{c,\mathrm{loc}}$ is normalized to start at zero.
The \b[<usual conditions for a filtration>] are completeness and right continuity. Completeness means that $\mathcal F_0$ contains every subset of each $\mathbb P$-null set in the ambient $\mathcal F$; right continuity means $\mathcal F_t=\bigcap_{u>t}\mathcal F_u$. These conditions concern the <filtered probability space>, whereas continuity of a <martingale> concerns its sample paths.
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