  Let $D$ be a bounded strongly convex domain with smooth boundary in $\mathbb
C^N$. Let $(\phi_t)$ be a continuous semigroup of holomorphic self-maps of $D$.
We prove that if $p\in \partial D$ is an isolated boundary regular fixed point
for $\phi_{t_0}$ for some $t_0>0$, then $p$ is a boundary regular fixed point
for $\phi_t$ for all $t\geq 0$. Along the way we also study backward iteration
sequences for elliptic holomorphic self-maps of $D$.
