Negation, Restriction, and Separated Differences of Semicontinuous Functions

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $A\subseteq X$ and $B\subseteq Y$, and let $\mathbb{R}$ be the ordered field of real numbers.

Let $u,w:A\to\mathbb{R}$, let $-u:A\to\mathbb{R}$ be the function whose value at $z\in A$ is $-u(z)$, and let $u-w:A\to\mathbb{R}$ be the function whose value at $z\in A$ is $u(z)-w(z)$. Then the following hold.

1. (Negation) Let $x\in A$. Then $u$ is upper semicontinuous at $x$ relative to $A$ if and only if $-u$ is lower semicontinuous at $x$ relative to $A$; and $u$ is lower semicontinuous at $x$ relative to $A$ if and only if $-u$ is upper semicontinuous at $x$ relative to $A$.

2. (Restriction) Let $A'\subseteq A$, let $x\in A'$, and let $u|_{A'}:A'\to\mathbb{R}$ be the function whose value at $z\in A'$ is $u(z)$. If $u$ is upper semicontinuous at $x$ relative to $A$, then $u|_{A'}$ is upper semicontinuous at $x$ relative to $A'$. If $u$ is lower semicontinuous at $x$ relative to $A$, then $u|_{A'}$ is lower semicontinuous at $x$ relative to $A'$.

3. (Difference) Let $x\in A$. If $u$ is upper semicontinuous at $x$ relative to $A$ and $w$ is lower semicontinuous at $x$ relative to $A$, then $u-w$ is upper semicontinuous at $x$ relative to $A$. If $u$ is lower semicontinuous at $x$ relative to $A$ and $w$ is upper semicontinuous at $x$ relative to $A$, then $u-w$ is lower semicontinuous at $x$ relative to $A$.

4. (Separated difference) Equip $X\times Y$ with the product metric $d_{X\times Y}$ obtained from $d_X$ and $d_Y$, which is a metric by claim 1 of The Product Metric is a Metric, and regard $A\times B$ as a subset of $X\times Y$. Let $f:A\to\mathbb{R}$ be upper semicontinuous on $A$, let $g:B\to\mathbb{R}$ be lower semicontinuous on $B$, and let $h:A\times B\to\mathbb{R}$ be the function whose value at $(x,y)\in A\times B$ is

$$h(x,y)=f(x)-g(y).$$

Then $h$ is upper semicontinuous on $A\times B$ with respect to the metric $d_{X\times Y}$.