Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Functions

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d_{X})$ be a metric space, let $A\subseteq X$, let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in the set of functions from $A$ to $\mathbb{R}$, and let $f:A\to\mathbb{R}$. The codomain $\mathbb{R}$ carries the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line, in which $d_{\mathbb{R}}(s,t)=|s-t|$, and continuity at a point relative to $A$, continuity on $A$ and uniform continuity on $A$ are those notions for maps from $A$ to $\mathbb{R}$, formed with $d_{X}$ and $d_{\mathbb{R}}$. Pointwise convergence and uniform convergence on a subset are as defined there, read with $D=A$. Then the following hold.

1. (Passing to a smaller set)¶ Let $T\subseteq S\subseteq A$ and suppose that $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $S$. Then $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $T$, and converges pointwise to $f$ on $S$.

2. (Continuity at a point from uniform convergence near it)¶ Let $x\in A$, let $r$ be a real number with $0<r$, and let $S\subseteq A$ be a set containing every point of the open ball $B_{d_{X}}(x,r)$ that lies in $A$. Suppose that $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $S$ and that $f_{k}$ is continuous at $x$ relative to $A$ for every $k\in\mathbb{N}$. Then $f$ is continuous at $x$ relative to $A$.

3. (Continuity on the whole set)¶ Suppose that $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $A$ and that $f_{k}$ is continuous on $A$ for every $k\in\mathbb{N}$. Then $f$ is continuous on $A$.

4. (Uniform continuity)¶ Suppose that $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $A$ and that $f_{k}$ is uniformly continuous on $A$ for every $k\in\mathbb{N}$. Then $f$ is uniformly continuous on $A$.