  In this paper, we consider a generalized two component Camassa-Holm system.
Based on local well-posedness results and lifespan estimates, we establish
sharpness of continuity on the data-to-solution map by showing that it is not
uniformly continuous from product Sobolev spaces $H^s \times H^{s}$ to
$C([0,T]; H^s \times H^{s})$. The proof of nonuniform dependence is based upon
approximate solutions.
