  We consider the Cauchy problem of the modified KdV equation (mKdV). Local
well-posedness of this problem is obtained in modulation spaces
$M^{1/4}_{2,q}(\mathbb{{R}})$ $(2\leq q\leq\infty)$. Moreover, we show that the
data-to-solution map fails to be $C^3$ continuous in
$M^{s}_{2,q}(\mathbb{{R}})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a
critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\geq
1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s'}$ with
$s'<1/4$. Noticing that $M^{1/4}_{2,q} \subset B^{1/q-1/4}_{2,q}$ is a sharp
embedding and $H^{-1/4}\subset B^{-1/4}_{2,\infty}$, our results contains all
of the subcritical data in $M^{1/4}_{2,q}$, which contains a class of functions
in $H^{-1/4}\setminus H^{1/4}$.
