Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $A\subseteq X$ and let $u:A\to\mathbb{R}$. Superlevel sets and the property of having closed superlevel sets in $X$ are as defined there, and a sequence in $X$ is said to converge when it converges in $(X,d)$. Then the following hold.

1. (Sequential characterisation)¶ The function $u$ has closed superlevel sets in $X$ if and only if the following condition holds: whenever $(x_{m})_{m\in\mathbb{N}}$ is a sequence in $A$ converging to a point $x\in X$ and $t\in\mathbb{R}$ satisfies $t\le u(x_{m})$ for every $m\in\mathbb{N}$, one has $x\in A$ and $t\le u(x)$.

2. (Upper semicontinuity)¶ If $u$ has closed superlevel sets in $X$, then $u$ is upper semicontinuous on $A$.

3. (Subtracting a continuous function)¶ Suppose $u$ has closed superlevel sets in $X$ and let $g:X\to\mathbb{R}$ be continuous on $X$, the metric on $\mathbb{R}$ being $d_{\mathbb{R}}$. Then the function $A\to\mathbb{R}$ whose value at $x\in A$ is $u(x)-g(x)$ has closed superlevel sets in $X$.

4. (Limits of values)¶ Suppose $u$ has closed superlevel sets in $X$, let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $A$ converging to a point $x\in X$, and suppose that the sequence $(u(x_{m}))_{m\in\mathbb{N}}$ converges to $L\in\mathbb{R}$. Then $x\in A$ and $L\le u(x)$.