  It is shown in \cite[Adv. Differ. Equ(2017)]{HT} that the Cauchy problem for
the generalized Camassa-Holm equation is well-posed in $C^1$ and the
data-to-solution map is H\"{o}lder continuous from $C^\alpha$ to
$\mathcal{C}([0,T];C^\alpha)$ with $\alpha\in[0,1)$. In this paper, we further
show that the data-to-solution map of the generalized Camassa-Holm equation is
not uniformly continuous on the initial data in $C^1$. In particular, our
result also can be a complement of previous work on the classical Camassa-Holm
equation in \cite[Geom. Funct. Anal(2002)]{G02}.
