  In the paper, we gave a strengthening of our previous work in [32] (J.
Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the
Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with
$s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method
applies also to the b-family of equations which contain the Camassa-Holm and
Degasperis-Procesi equations.
