  We solve an optimal stopping problem where the underlying diffusion is
Brownian motion on $\bf R$ with a positive drift changing at zero. It is
assumed that the drift $\mu_1$ on the negative side is smaller than the drift
$\mu_2$ on the positive side. The main observation is that if $\mu_2-\mu_1>1/2$
then there exists values of the discounting parameter for which it is not
optimal to stop in the vicinity of zero where the drift changes. However, when
the discounting gets bigger the stopping region becomes connected and contains
zero. This is in contrast with results concerning optimal stopping of skew
Brownian motion where the skew point is for all values of the discounting
parameter in the continuation region.
