Dilepton production rate (DPR) from hot and dense quark matter is studied in the presence of an arbitrary external magnetic field using the 2-flavour Nambu--Jona-Lasinio (NJL) model. The anomalous magnetic moment (AMM) of the quarks is taken into consideration while calculating the constituent quark mass as well as the DPR from the thermo-magnetic medium. An infinite number of quark Landau levels is incorporated so that no approximations are made on the strength of the background magnetic field. The analytic structure of the two point vector current correlation function in the complex energy plane reveals that, in addition to the usual Unitary cut, a non-trival Landau cut appears in the physical kinematic domains solely due to the external magnetic field. Moreover, these kinematic domains of the Unitary and Landau cuts are found to be significantly modified due to the AMM of the quarks. With finite AMM of the quarks, for certain values of the external magnetic field, the kinematically forbidden gap between the Unitary and Landau cuts are shown to vanish leading to the generation of a continuous spectrum of dilepton emission over the whole invariant mass region not observed earlier.
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