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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.

1. 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.

2. 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.

Addendum 1

I made question 1 more general than I needed it to be, and in the process introduced a few mistakes, which Alex Becker and Robert Israel helped me see. I am reformulating statement 1 as follows.

Let $X$ be a compact metric space and $Y$ and $Z$ be topological spaces. Let $f : X \times Y \rightarrow Z$ be a continuous map. Let the map $g : Y \times Z \rightarrow \mathcal{P}(X)$ be given by $g(y,z) = (f(\cdot,y))^{-1}(z) = \{ x \in X : f(x,y) = z \}$. Let $z \in Z$ such that $g(y,z) \neq \varnothing$ for all $y \in Y$. The map $g(\cdot,z) : Y \rightarrow \mathcal{H}(X)$ is (well-defined and) continuous.

I probably need the spaces $X$, $Y$, $Z$ not to have pathologies, e.g. multiple connected components; maybe just convex in Euclidean space to get started.

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.

1. 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.

2. 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.

Addendum 1

I made question 1 more general than I needed it to be, and in the process introduced a few mistakes, which Alex Becker and Robert Israel helped me see. I am reformulating statement 1 as follows.

Let $X$ be a compact metric space and $Y$ and $Z$ be topological spaces. Let $f : X \times Y \rightarrow Z$ be a continuous map. Let the map $g : Y \times Z \rightarrow \mathcal{P}(X)$ be given by $g(y,z) = (f(\cdot,y))^{-1}(z) = \{ x \in X : f(x,y) = z \}$. Let $z \in Z$ such that $g(y,z) \neq \varnothing$ for all $y \in Y$. The map $g(\cdot,z) : Y \rightarrow \mathcal{H}(X)$ is (well-defined and) continuous.

I probably need the spaces $X$, $Y$, $Z$ not to have pathologies, e.g. multiple connected components; maybe just convex in Euclidean space to get started.

An intuition for this problem is as follow. Consider that $f(x,y) = z$ is a solvable equation where $z$ is fixed, $y$ is a parameter ranging over $Y$, and $x$ is the unknown. I would like to say that the solution set, i.e. $g(y,z)$, depends continuously on the parameter $y$.

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.

1. 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.

2. 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.

Addendum 1

I made question 1 more general than I needed it to be, and in the process introduced a few mistakes, which Alex Becker and Robert Israel helped me see. I am reformulating statement 1 as follows.

Let $X$ be a compact metric space and $Y$ and $Z$ be topological spaces. Let $f : X \times Y \rightarrow Z$ be a continuous map. Let the map $g : Y \times Z \rightarrow \mathcal{P}(X)$ be given by $g(y,z) = (f(\cdot,y))^{-1}(z) = \{ x \in X : f(x,y) = z \}$. Let $z \in Z$ such that $g(y,z) \neq \varnothing$ for all $y \in Y$. The map $g(\cdot,z) : Y \rightarrow \mathcal{H}(X)$ is (well-defined and) continuous.

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.

1. 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.

2. 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.

Addendum 1

I made question 1 more general than I needed it to be, and in the process introduced a few mistakes, which Alex Becker and Robert Israel helped me see. I am reformulating statement 1 as follows.

Let $X$ be a compact metric space and $Y$ and $Z$ be topological spaces. Let $f : X \times Y \rightarrow Z$ be a continuous map. Let the map $g : Y \times Z \rightarrow \mathcal{P}(X)$ be given by $g(y,z) = (f(\cdot,y))^{-1}(z) = \{ x \in X : f(x,y) = z \}$. Let $z \in Z$ such that $g(y,z) \neq \varnothing$ for all $y \in Y$. The map $g(\cdot,z) : Y \rightarrow \mathcal{H}(X)$ is (well-defined and) continuous.

I probably need the spaces $X$, $Y$, $Z$ not to have pathologies, e.g. multiple connected components; maybe just convex in Euclidean space to get started.

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.

1. 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.

2. 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.

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.

1. 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.

2. 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.

Addendum 1

I made question 1 more general than I needed it to be, and in the process introduced a few mistakes, which Alex Becker and Robert Israel helped me see. I am reformulating statement 1 as follows.

Let $X$ be a compact metric space and $Y$ and $Z$ be topological spaces. Let $f : X \times Y \rightarrow Z$ be a continuous map. Let the map $g : Y \times Z \rightarrow \mathcal{P}(X)$ be given by $g(y,z) = (f(\cdot,y))^{-1}(z) = \{ x \in X : f(x,y) = z \}$. Let $z \in Z$ such that $g(y,z) \neq \varnothing$ for all $y \in Y$. The map $g(\cdot,z) : Y \rightarrow \mathcal{H}(X)$ is (well-defined and) continuous.