The necessary and sufficient condition for consistency of the MLE

Suppose that the observations are i.i.d. from a density f(.;^θ), where θ is an identifiable parameter. One expects that the maximum likelihood estimator of θ is consistent. But its consistency proof is non-trivial and various sufficient conditions have been proposed (see, e.g., the classical statistics textbooks). All these sufficient conditions require _f(x_{; θ}) being somewhat upper semi-continuous (in θ), with various smoothness conditions or conditions needed for the dominated convergence theorem. We study the sufficient and necessary condition.