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A novel resource,i.e.measurement-induced nonlocality,was proposed by Luo and Fu [Phys.Rev.Lett.106(2011)120401].It can directly reflect the usefulness of quantum nonlocality in quantum protocols.We investigate the dynamics of measurement-induced nonlocality by exactly solving a model which consists of two independent atoms each subject to a zero-temperature muti-mode cavity.We find that the dynamics of measurement-induced nonlocality is discontinuity under some special conditions.The reason for this phenomenon is the constraint of the von Neumann measurements which do not disturb the reduced density operator.We also find the sudden death phenomenon and the sudden birth phenomenon.