We investigate how a principal's knowledge of agents' higher-order beliefs impacts his ability to robustly implement a given social choice function. We adapt a formulation of Oury and Terceiux (2012): a social choice function is continuously implementable if it is partially implementable for types in an initial model (here, common knowledge of preferences) and "nearby" types. We characterize when a social choice function is truthfully continuously implementable, i.e., using game forms corresponding to direct revelation mechanisms for the initial model. Our characterization hinges on how our formalization of the notion of nearby preserves agents' higher order beliefs. If nearby types have similar higher order beliefs, truthful continuous implementation is roughly equivalent to requiring that the social choice function is implementable in Strict Nash equilibrium in the initial model, a very permissive solution concept. If they do not, then our notion is equivalent to requiring that the social choice function is implementable in rationalizable strategies in the initial model, and further that there be a unique rationalizable strategy. This is a very restrictive requirement. If only ordinal preferences are common knowledge among agents, a mild richnesscondition implies that the social choice function must be dictatorial.Truthful continuous implementation is thus impossible without non-trivial knowledge of agents' higher order beliefs.