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The principle of smooth fit is probably the most used tool to find solutionsto optimal stopping problems of one-dimensional diffusions. It is important,e.g., in financial mathematical applications to understand in which kind ofmodels and problems smooth fit can fail. In this paper we connect - in case ofone-dimensional diffusions - the validity of smooth fit and thedifferentiability of excessive functions. The basic tool to derive the resultsis the representation theory of excessive functions; in particular, the Rieszand Martin representations. It is seen that the differentiability may not holdin case the speed measure of the diffusion or the representing measure of theexcessive function has atoms.  As an example, we study optimal stopping of sticky Brownian motion. It isknown that the validity of the smooth fit in this case depends on the value ofthe discounting parameter (when the other parameters are fixed). We decomposethe size of the jump in the derivative of the value function in two factors.The first one is due to the atom of the representing measure and the second onedue to the atom of the speed measure.

Differentiablity of excessive functions of one-dimensional diffusions and the principle of smooth fit