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 Tercieux (2012): a social choice function is continuously implementable if it is partially implementable for types in an initial model 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 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 unique rationalizable strategies in the initial model. If only ordinal preferences are common knowledge among agents, a mild richness condition implies that the social choice function must be dictatorial. Truthful continuous implementation is thus impossible without non-trivial knowledge of agents' higher order beliefs. We further show that without such knowledge, a revelation principle does not apply: the set of social choice functions which can be continuously implemented is strictly larger.