Lower Semicontinuous Function on a Subset of a Metric Space

Let $(X,d)$ be a metric space, let $A\subseteq X$, 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 $a\ne b$, let $u:A\to\mathbb{R}$, and let $x\in A$.

We say that $u$ is lower semicontinuous at $x$ relative to $A$ if for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in A$ satisfying $d(x,y)<\delta$ satisfies

where $u(x)-\varepsilon$ abbreviates $u(x)+(-\varepsilon)$.

We say that $u$ is lower semicontinuous on $A$ if $u$ is lower semicontinuous at $x$ relative to $A$ for every $x\in A$.