Before revision:  We also show that the exercise boundary is continuous and a high contact principle holds in such a neighbourhood when the dividend function is moreover linearin a neighbourhood of 0.
After revision:  We also show that the exercise boundary is continuous and a high contact principle in the left-hand neighbourhood of t_d. When it is globally linear, then right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.