The problem is that the continuity condition seems to be a dummy condition. Here is my proof.
Let $X$ be in $\mathscr{M}_2^c$ (Continuous square-integrable martingale and $X_0=0$), and $T$ be a stopping time of $\mathscr{F}_t$. If $\langle X\rangle_T=0$ a.s. $P$, then we have $P[X_{T\land t}=0, \forall 0\leq t<\infty]=1$
Proof： For $\langle X\rangle_t$ is non-negative and non-decreasing, we have $0\leq \langle X\rangle_{T\land t}\leq \langle X\rangle_T=0$ for all $t$. By the definition of $\langle X\rangle_t$, $X_t^2-\langle X\rangle_t$ is a right-continuous martingale, which implies $X_{T\land t}^2-\langle X\rangle_{T\land t}$ is an $R$-martingale. \begin{align} E[X_{T\land t}^2]=E\langle X\rangle_{T\land t}=0 ,\qquad \forall 0\leq t<\infty \end{align} i.e. for all $0\leq t<\infty$, $X_{T\land t}=0$ a.s. $P$. Since $0$ is a modification of $X_{T\land t}$ and $X_{T\land t}$ is right-continuous, $X_{T\land t}$ and $0$ are indistinguishable. i.e. $P[X_{T\land t}=0, \forall 0\leq t<\infty]=1$.
Does anyone have any comment? Thanks in advance!