Berge's theorem

Berge's theorem states that if you have a set of options (like choices in a problem) that change smoothly with some parameters, then the best options (maximizers or minimizers) also change smoothly, and you can track their behavior consistently. Specifically, if the set of feasible choices depends continuously and the objective function is well-behaved (upper or lower semi-continuous), then the optimal solutions vary continuously with the parameters. This theorem ensures stability: small changes in conditions lead to small changes in the optimal solutions, making it valuable in optimization and economic modeling.