  This paper introduces a version of the argmax continuous mapping theorem that
applies to M-estimation problems in which the objective functions converge to a
limiting process with multiple maximizers. The concept of the smallest
maximizer of a function in the d-dimensional Skorohod space is introduced and
its main properties are studied. The resulting continuous mapping theorem is
applied to three problems arising in change-point regression analysis. Some of
the results proved in connection to the d-dimensional Skorohod space are also
of independent interest.
