  In this paper we consider the hyperelastic rod equation on the Sobolev spaces
$H^s(\R)$, $s > 3/2$. Using a geometric approach we show that for any $T > 0$
the corresponding solution map, $u(0) \mapsto u(T)$, is nowhere locally
uniformly continuous. The method applies also to the periodic case $H^s(\mathbb
T)$, $s > 3/2$.
