Is the Legendre transformation continuous on the space of the convex $\mathbb R^{n} \to \mathbb R$ functions for the topology of the simple convergence?
First the limit of a sequence of convex function is a convex function in regard of the simple convergence. And it's easy to see that
$$u^{*}(x) \le \liminf_{n} u_{n}^{*}$$
So can we conclude?