Let $X \subset \mathbb{R}^n$ be a finite set and $\Phi : X \mapsto 2^{X}$ be a set-valued mapping defined as follows:
$\Phi(y) := \underset{x \in X}{\textrm{argmin}} \; L(x, y)$.
I'm trying to figure out if, without any particular assumptions on $L: X \times X \to \mathbb{R}$, $\Phi(\cdot)$ is continuous over the finite set $X$. I guess no - any hints or suggestions?