  It is proved that if $u_0\in B^s_{p,r}$ with $s>1+\frac1p,
(p,r)\in[1,+\infty]\times[1,+\infty)$ or $s=1+\frac1p, \
(p,r)\in[1,+\infty)\times \{1\}$, the solution of the Camassa--Holm equation
belongs to $\mathcal{C}([0,T];B^s_{p,r})$. In the paper, we show that the
continuity of the solution can not be improved to the H\"{o}lder continuity.
Precisely speaking, the solution of the Camassa--Holm equation belongs to
$\mathcal{C}([0,T];B^s_{p,r})$ but not to $\mathcal{C}^\alpha([0,T];B^s_{p,r})$
with any $\alpha\in(0,1)$.
