Conditions for Limit Function to be Limit Minimizing Function of Functional

Theorem
Let $ y $ be a real function.

Let $ J \left [ { y } \right ] $ be a functional.

Let $ \left \{ { y_n } \right \} $ be a minimizing sequence of $ J $.

Let:


 * $ \displaystyle \lim_{ n \to \infty } y_n = \hat{ y } $

Suppose $ J $ is semicontinuous at $ y = \hat{ y } $.

Then:


 * $ \displaystyle J \left [ { \hat{ y } } \right ] = \lim_{ n \to \infty } J \left [ { y_n } \right ] $