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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 equationbelongs to $\mathcal{C}([0,T];B^s_{p,r})$. In the paper, we show that thecontinuity 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 $\alpha\in(0,1)$.

The failure of H\"{o}lder regularity of solutions for the Camassa--Holm type equation in Besov spaces