  In this paper, we study the Cauchy problem for the Camassa-Holm equation on
the real line. By presenting a new construction of initial data, we show that
the solution map in the smaller space $B_{p,1}^{1}\cap C^{0,1}$ with
$p\in(2,\infty]$ is discontinuous at origin. More precisely, $u_0\in
B_{p,1}^{1}\cap C^{0,1}$ can guarantee that the Camassa-Holm equation has a
unique local solution in $W^{1,p}\cap C^{0,1}$, however, this solution is
instable and can have an inflation in $B_{p,1}^{1}\cap C^{0,1}$ for certain
initial data.
