Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set

In the setting of The Real Numbers: Standing Notation and Background, let $(M,d)$ be a metric space, let $S\subseteq M$ be nonempty, let $O\subseteq M$ be open in $(M,d)$, and suppose that the set $S'=S\cap O$ is nonempty. For a function $u:S\to\mathbb{R}$ and a nonempty $B\subseteq S$, $u|_{B}:B\to\mathbb{R}$ denotes the function whose value at $z\in B$ is $u(z)$.

That a function on $S$, or on $S'$, is bounded above near each point, or bounded below near each point, of that set, and the resulting upper semicontinuous envelope $(\cdot)^{*}$ and lower semicontinuous envelope $(\cdot)_{*}$, are understood with the ambient metric space $(M,d)$; so are local maxima and local minima relative to that set.

Let $u:S\to\mathbb{R}$. Then the following hold.

1. (Local bounds restrict)(i) If $u$ is bounded above near each point of $S$, then $u|_{S'}$ is bounded above near each point of $S'$; if $u$ is bounded below near each point of $S$, then $u|_{S'}$ is bounded below near each point of $S'$.

2. (Locality of the upper envelope)(ii) Suppose $u$ is bounded above near each point of $S$. Then

$$\bigl(u|_{S'}\bigr)^{*}(x)=u^{*}(x)\qquad\text{for every }x\in S' .$$

3. (Locality of the lower envelope)(iii) Suppose $u$ is bounded below near each point of $S$. Then

$$\bigl(u|_{S'}\bigr)_{*}(x)=u_{*}(x)\qquad\text{for every }x\in S' .$$

4. (Locality of local maxima)(iv) Let $\bar{x}\in S'$. Then $u$ has a local maximum at $\bar{x}$ relative to $S$ if and only if $u|_{S'}$ has a local maximum at $\bar{x}$ relative to $S'$.

5. (Locality of local minima)(v) Let $\bar{x}\in S'$. Then $u$ has a local minimum at $\bar{x}$ relative to $S$ if and only if $u|_{S'}$ has a local minimum at $\bar{x}$ relative to $S'$.