Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space

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. 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 where $s-t$ abbreviates $s+(-t)$. Let $\mathbb{N}$ be the set of natural numbers. Let $u:A\to\mathbb{R}$ and let $x\in A$.

Then the following hold.

1. (Sequential characterization at a point.) $u$ is lower semicontinuous at $x$ relative to $A$ if and only if the following condition holds: for every sequence $(y_{j})_{j\in\mathbb{N}}$ in $A$ that converges to $x$ in the metric space $(A,d_{A})$, and for every real $\varepsilon>0$, there is $N\in\mathbb{N}$ such that

$$u(x)-\varepsilon<u(y_{j})\qquad\text{for every }j\in\mathbb{N}\text{ with }j\ge N .$$

2. (Limit inferior form.) Assume that $u$ is lower semicontinuous at $x$ relative to $A$, let $(y_{j})_{j\in\mathbb{N}}$ be a sequence in $A$ converging to $x$ in $(A,d_{A})$, and assume that the real sequence $\bigl(u(y_{j})\bigr)_{j\in\mathbb{N}}$ is bounded, so that its limit inferior is defined. Then

$$u(x)\le\liminf_{j}u(y_{j}).$$

3. (Sequential characterization on a subset.) $u$ is lower semicontinuous on $A$ if and only if the condition stated in claim 1 holds at every $x\in A$.