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In the present paper, we investigate the complex dynamics arising from a behavioral exchange rate discontinuous model with heterogeneous agents. Unlike previous works explaining the emergence of chaos in the exchange rate models as the resulting of nonlinearity, our model is able to produce endogenous exchange rate dynamics due to the presence of discontinuity induced by a sentiment index, which affects the way investors take their trading decisions. In particular, it affects the level of optimism/pessimism of fundamentalists regarding their perception on the value of fundamental. Moreover, it also affects the strength to which one kind of chartists places her buying/selling orders. We show that our model, represented by a two-dimensional discontinuous map, has the ability to produce interesting endogenous exchange rate dynamics. In addition, when each component of the map is buffeted by a stochastic component, the model closely replicates the stylized facts of the EUR/USD and EUR/JPY exchange rate markets. [ABSTRACT FROM AUTHOR]