  We provide sufficient conditions for the continuity of the free-boundary in a
general class of finite-horizon optimal stopping problems arising for instance
in finance and economics. The underlying process is a strong solution of one
dimensional, time-homogeneous stochastic differential equation (SDE). The proof
relies on both analytic and probabilistic arguments and it is based on a
contradiction scheme inspired by the maximum principle in partial differential
equations (PDE) theory. Mild, local regularity of the coefficients of the SDE
and smoothness of the gain function locally at the boundary are required.
