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

A note on the continuity of free-boundaries in finite-horizon optimal stopping problems for one dimensional diffusions