Proof of Semicontinuity and Continuity Under Composition with a Continuous Map

Fix $y\in B$ and a real number $\varepsilon$ with $0<\varepsilon$; the point $g(y)$ lies in $A$. Since $g$ is continuous at $y$ relative to $B$, for every real number $\eta$ with $0<\eta$ there is a real number $\delta$ with $0<\delta$ such that every $y'\in B$ with $d_Y(y,y')<\delta$ satisfies $d_X(g(y'),g(y))<\eta$, hence also $d_X(g(y),g(y'))<\eta$ by condition 3 in the definition of a metric. We use this repeatedly.

Claim 1. Suppose $u$ is upper semicontinuous at $g(y)$ relative to $A$. Choose $\eta$ with $0<\eta$ such that every $a\in A$ with $d_X(g(y),a)<\eta$ satisfies $u(a)<u(g(y))+\varepsilon$, and choose $\delta$ for this $\eta$ as above. If $y'\in B$ and $d_Y(y,y')<\delta$, then $g(y')\in A$ and $d_X(g(y),g(y'))<\eta$, so

$$(u\circ g)(y')=u(g(y'))<u(g(y))+\varepsilon=(u\circ g)(y)+\varepsilon .$$

Hence $u\circ g$ is upper semicontinuous at $y$ relative to $B$. As $y\in B$ was arbitrary, $u\circ g$ is upper semicontinuous on $B$.

Claim 2. Suppose $u$ is lower semicontinuous at $g(y)$ relative to $A$. Choose $\eta$ with $0<\eta$ such that every $a\in A$ with $d_X(g(y),a)<\eta$ satisfies $u(g(y))-\varepsilon<u(a)$, and choose $\delta$ for this $\eta$. If $y'\in B$ and $d_Y(y,y')<\delta$, then as before $(u\circ g)(y)-\varepsilon<(u\circ g)(y')$. Hence $u\circ g$ is lower semicontinuous at $y$ relative to $B$, and therefore on $B$.

Claim 3. Suppose $u$ is continuous at $g(y)$ relative to $A$. Choose $\eta$ with $0<\eta$ such that every $a\in A$ with $d_X(g(y),a)<\eta$ satisfies $d_{\mathbb{R}}(u(a),u(g(y)))<\varepsilon$, and choose $\delta$ for this $\eta$. If $y'\in B$ and $d_Y(y,y')<\delta$, then $d_{\mathbb{R}}((u\circ g)(y'),(u\circ g)(y))<\varepsilon$. Hence $u\circ g$ is continuous at $y$ relative to $B$, and therefore on $B$.

Claim 4. Apply claims 1, 2 and 3 with $(Y,d_Y)$ taken to be $(X,d_X)$, with $B$ taken to be $C$, and with $g:C\to X$ the map $g(c)=c$. This map satisfies $g(c)\in A$ for every $c\in C$ because $C\subseteq A$, and it is continuous on $C$ relative to $C$: given $c\in C$ and a real number $\varepsilon$ with $0<\varepsilon$, take $\delta=\varepsilon$, and note that $c'\in C$ with $d_X(c,c')<\delta$ satisfies $d_X(g(c'),g(c))=d_X(c',c)=d_X(c,c')<\varepsilon$ by condition 3 in the definition of a metric. Since $u\circ g=u|_{C}$, the three assertions of claim 4 follow.