Consider a continuous function $f: X \times U \to \mathbb{R}$ satisfying $f(x, \cdot)$ is convex for any $x \in X$ where $X \subset \mathbb{R}^{n}$ is compact and $U \subset \mathbb{R}^{m}$ is convex compact.
Also, consider the minimizer function $g(x) := \arg \min_{u \in U} f(x, u)$. Note that $g(x)$ can be a set, not a single value.
Is the graph of $g$ closed? In other words, if $x_{i} \to x$, $u^{*}_{i} \to u^{*} $ where $x_{i} \in X$ and $u_{i}^{*} \in \arg\min_{u \in U} f(x_i, u)$, then $u^{*} \in \arg \min_{u \in U} f(x, u)$?