For a metric space $E$, let $\mathcal{H}(E)$ be the metric space consisting of the set of nonempty compact subsets of $E$ and the Hausdorff metric. Consider the following two statements.
Let $X$ and $Z$ be topological spaces and let $Y$ be a compact metric space. Let $f : X \times Y \rightarrow Z$ be a continuous map. Let the map $g : Y \times Z \rightarrow \mathcal{H}(X)$ be given by $g(y,z) = (f(y,\cdot))^{-1}(z) = \{ x \in X : f(x,y) = z \}$. The map $g$ is continuous.
The maps $\min : \mathcal{H}(\mathbb{R}) \rightarrow \mathbb{R}$ and $\max : \mathcal{H}(\mathbb{R}) \rightarrow \mathbb{R}$ are continuous.
I am looking for references to literature where these statements (which do sound true) can be found. Much thanks in advance.