Let $f$ be merely continuous (not necessarily bounded). Then either the CDE $dy_t = f(y_t)\,dx_t$ admits a global solution on $[0,T]$, or there exists a maximal time $\tau \in (0,T]$ such that $\lim_{t \nearrow \tau} \|y_t\| = +\infty$.
(Friz–Victoir 2010, Theorem 3.6)