Suppose $X \subseteq \mathbb{R}^m$ and $Y \subseteq \mathbb{R}^n$. In addition, assume $Y$ is a bounded, complete lattice with respect to the product order. Suppose $\Gamma: X \to 2^Y$ is a correspondence from $X$ to subsets of $Y$ such that $\Gamma(x) \subset Y$ is a complete sublattice for all $x \in X$.
If $\Gamma(\cdot)$ is continuous, is $\bar{g}(x) := \sup \Gamma(x)$ also continuous?