Local Minimum of a Function Relative to 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 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 $u$ has a local minimum at $x$ relative to $A$ if there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in A$ satisfying $d(x,y)<\delta$ satisfies

We say that $u$ has a strict local minimum at $x$ relative to $A$ if there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in A$ satisfying $d(x,y)<\delta$ and $y\ne x$ satisfies