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 a one-dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the boundary are required.