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A continuity structure of correlations among arms in multi-armed bandit can bring a significant acceleration of exploration and reduction of regret, in particular, when there are many arms. However, it is often latent in practice. To cope with the latent continuity, we consider a transfer learning setting where an agent learns the structural information, parameterized by a Lipschitz constant and an embedding of arms, from a sequence of past tasks and transfers it to a new one. We propose a simple but provably-efficient algorithm to accurately estimate and fully exploit the Lipschitz continuity at the same asymptotic order of lower bound of sample complexity in the previous tasks. The proposed algorithm is applicable to estimate not only a latent Lipschitz constant given an embedding, but also a latent embedding, while the latter requires slightly more sample complexity. To be specific, we analyze the efficiency of the proposed framework in two folds: (i) our regret bound on the new task is close to that of the oracle algorithm with the full knowledge of the Lipschitz continuity under mild assumptions; and (ii) the sample complexity of our estimator matches with the information-theoretic fundamental limit. Our analysis reveals a set of useful insights on transfer learning for latent Lipschitz continuity. From a numerical evaluation based on real-world dataset of rate adaptation in time-varying wireless channel, we demonstrate the theoretical findings and show the superiority of the proposed framework compared to baselines.