Before revision:  My result implies that small perturbations to a pure commitment action can lead to a discontinuous change in the patient player's equilibrium payoff. The main technical contribution is to use martingale techniques to construct a non-stationary strategy under which the patient player can avoid signaling negative information about the state while at the same time, matching the long-run frequency of his actions to the mixed commitment action and convincing his opponents that his action is close to the commitment action in almost all periods .
After revision:  My result implies that small perturbations to a pure commitment action can lead to a discontinuous change in the patient player's guaranteed equilibrium payoff .