Consider a locally-bounded, continuous, positive-semidefinite function $f: X \times Y \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact, $Y \subseteq \mathbb{R}^m$.
For each $y \in Y$, define:
$ f^*(y) := \min_{x \in X} f(x,y) $
$ x^*(y) := \arg\min_{x \in X} f(x,y) $
Assume that the optimal & optimizer always exist and are finite.
It is known that, under these conditions, $f^*$ is a continuous function.
1) Provide an example in which $x^*$ is not continuous (while $f^*$ necessarily does).
2) Provide an example in which $x^*$ is not continuous outside the origin.