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We prove that the smallest minimizer s(f) of a real convex function f is lessthan or equal to a real point x if and only if the right derivative of f at xis non-negative. Similarly, the largest minimizer t(f) is greater or equal to xif and only if the left derivative of f at x is non-positive. From this simpleresult we deduce measurability and semi-continuity of the functionals s and t.Furthermore, if f has a unique minimizing point, so that s(f) = t(f), then thefunctional is continuous at f. With these analytical preparations we can applyContinuous Mapping Theorems to obtain several Argmin theorems for convexstochastic processes. The novelty here are statements about classicaldistributional convergence and almost sure convergence, if the limit processdoes not have a unique minimum point. This is possible by replacing the naturaltopology on R with the order topologies. Another new feature is that not onlysequences but more generally nets of convex stochastic processes are allowed.

On semi-continuity and continuity of the smallest and largest minimizing  point of real convex functions with applications in probability and  statistics