  The divide and color model on a graph $G$ arises by first deleting each edge
of $G$ with probability $1-p$ independently of each other, then coloring the
resulting connected components (\emph{i.e.}, every vertex in the component)
black or white with respective probabilities $r$ and $1-r$, independently for
different components. Viewing it as a (dependent) site percolation model, one
can define the critical point $r_c^G(p)$.
  In this paper, we mainly study the continuity properties of the function
$r_c^G$, which is an instance of the question of locality for percolation. Our
main result is the fact that in the case $G=\mathbb Z^2$, $r_c^G$ is continuous
on the interval $[0,1/2)$; we also prove continuity at $p=0$ for the more
general class of graphs with bounded degree. We then investigate the sharpness
of the bounded degree condition and the monotonicity of $r_c^G(p)$ as a
function of $p$.
