  We prove that the smallest minimizer s(f) of a real convex function f is less
than or equal to a real point x if and only if the right derivative of f at x
is non-negative. Similarly, the largest minimizer t(f) is greater or equal to x
if and only if the left derivative of f at x is non-positive. From this simple
result 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 the
functional is continuous at f. With these analytical preparations we can apply
Continuous Mapping Theorems to obtain several Argmin theorems for convex
stochastic processes. The novelty here are statements about classical
distributional convergence and almost sure convergence, if the limit process
does not have a unique minimum point. This is possible by replacing the natural
topology on R with the order topologies. Another new feature is that not only
sequences but more generally nets of convex stochastic processes are allowed.
