= Solution
The strictly increasing continuous clock $[M]$ has a finite continuous inverse $\tau_s$. The <time change of a continuous process> therefore preserves continuity and adaptedness. Moreover,
$$
\widetilde X_s
=X_0+M_{\tau_s}+A_{\tau_s}.
$$
The <optional time-change theorem> makes $M_{\tau_s}$ a continuous local martingale for $\widetilde{\mathcal F}_s=\mathcal F_{\tau_s}$, while composition with the increasing map $s\mapsto\tau_s$ preserves the finite variation of $A$. Hence $\widetilde X$ is a continuous semimartingale in the time-changed filtration.
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