Semicontinuity and Continuity Under Composition with a Continuous Map

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 set of real numbers with the addition and the order $\le$ of its ordered field structure, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Numbers.

Let $g:B\to X$ be continuous on $B$ relative to $B$ and satisfy $g(y)\in A$ for every $y\in B$, let $u:A\to\mathbb{R}$, and let $u\circ g:B\to\mathbb{R}$ be the function with $(u\circ g)(y)=u(g(y))$ for $y\in B$. Then the following hold.

1. If $u$ is upper semicontinuous on $A$, then $u\circ g$ is upper semicontinuous on $B$.

2. If $u$ is lower semicontinuous on $A$, then $u\circ g$ is lower semicontinuous on $B$.

3. If $u$ is continuous on $A$ relative to $A$, as a map into $(\mathbb{R},d_{\mathbb{R}})$, then $u\circ g$ is continuous on $B$ relative to $B$.

4. (Restriction) Let $C\subseteq A$ and let $u|_{C}:C\to\mathbb{R}$ be the function with $u|_{C}(c)=u(c)$ for $c\in C$. If $u$ is upper semicontinuous on $A$, then $u|_{C}$ is upper semicontinuous on $C$; if $u$ is lower semicontinuous on $A$, then $u|_{C}$ is lower semicontinuous on $C$; and if $u$ is continuous on $A$ relative to $A$, then $u|_{C}$ is continuous on $C$ relative to $C$.