Semicontinuity via Sublevel and Superlevel Sets

Let $(X,d)$ be a metric space, let $A\subseteq X$, and let $d_A$ be the restriction of $d$ to $A$, a metric on $A$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology. Equip $A$ with the collection of its subsets that are open in $(A,d_A)$, which is a topology by Metric Open Sets Form a Topology. Let $\mathbb{R}$ be the set of real numbers with the addition and the order $\le$ of its ordered field structure, where $s<t$ means that $s\le t$ and $s\ne t$, and let $u:A\to\mathbb{R}$. For $c\in\mathbb{R}$ set

$$A_{u<c}=\{y\in A: u(y)<c\},\qquad A_{c\le u}=\{y\in A: c\le u(y)\},$$ $$A_{c<u}=\{y\in A: c<u(y)\},\qquad A_{u\le c}=\{y\in A: u(y)\le c\}.$$

Then the following hold.

1. $u$ is upper semicontinuous on $A$ if and only if $A_{u<c}$ is open in $(A,d_A)$ for every $c\in\mathbb{R}$.

2. $u$ is lower semicontinuous on $A$ if and only if $A_{c<u}$ is open in $(A,d_A)$ for every $c\in\mathbb{R}$.

3. If $u$ is upper semicontinuous on $A$, then $A_{c\le u}$ is closed in the topological space $A$ for every $c\in\mathbb{R}$; if $u$ is lower semicontinuous on $A$, then $A_{u\le c}$ is closed in the topological space $A$ for every $c\in\mathbb{R}$.