Semicontinuity and the Semicontinuous Envelopes are Local Notions

Let $(M,d)$ be a metric space, let $S\subseteq M$ be nonempty, let $\mathbb{R}$ be the set of real numbers with the addition and the order $\le$ of its ordered field structure, where $a<b$ means that $a\le b$ and $a\ne b$ and $a-b$ abbreviates $a+(-b)$, and let $u:S\to\mathbb{R}$.

Let $x\in S$ and let $r\in\mathbb{R}$ be positive. Put

$$S_{r}=\{y\in S:d(y,x)\le r\},$$

which contains $x$, since $d(x,x)=0$ by the metric axioms and $0\le r$; in particular $S_{r}$ is nonempty. Let $u|_{S_{r}}:S_{r}\to\mathbb{R}$ be the function whose value at $z\in S_{r}$ is $u(z)$, and more generally, for $V\subseteq S$, let $u|_{V}$ denote the analogous restriction. That $u$ is bounded above near each point of $S$ or bounded below near each point of $S$, and the resulting upper semicontinuous envelope $u^{*}$ and lower semicontinuous envelope $u_{*}$, are as defined there.

Then the following hold.

1. (Locality of upper semicontinuity)(i) $u$ is upper semicontinuous at $x$ relative to $S$ if and only if $u|_{S_{r}}$ is upper semicontinuous at $x$ relative to $S_{r}$.

2. (Locality of lower semicontinuity)(ii) $u$ is lower semicontinuous at $x$ relative to $S$ if and only if $u|_{S_{r}}$ is lower semicontinuous at $x$ relative to $S_{r}$.

3. (Semicontinuity on a neighbourhood)(iii) Let $V\subseteq S$ satisfy $S_{r}\subseteq V$. If $u|_{V}$ is upper semicontinuous at $x$ relative to $V$, then $u$ is upper semicontinuous at $x$ relative to $S$; if $u|_{V}$ is lower semicontinuous at $x$ relative to $V$, then $u$ is lower semicontinuous at $x$ relative to $S$.

4. (Locality of the upper envelope)(iv) Suppose $u$ is bounded above near each point of $S$. Then $u|_{S_{r}}$ is bounded above near each point of $S_{r}$, and

$$\bigl(u|_{S_{r}}\bigr)^{*}(y)=u^{*}(y)\qquad\text{for every }y\in S\text{ with }d(y,x)<r .$$

5. (Locality of the lower envelope)(v) Suppose $u$ is bounded below near each point of $S$. Then $u|_{S_{r}}$ is bounded below near each point of $S_{r}$, and

$$\bigl(u|_{S_{r}}\bigr)_{*}(y)=u_{*}(y)\qquad\text{for every }y\in S\text{ with }d(y,x)<r .$$