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

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, 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 $\mathbb{R}^{A}$ of maps from $A$ to $\mathbb{R}$ of Sets and Maps: Ordinary Notation (v), and let $f:A\to\mathbb{R}$. For maps from $A$ to $\mathbb{R}$, continuity at a point relative to $A$ and continuity on $A$ are those of Continuous Map Between Metric Spaces, and uniform continuity on $A$ that of Uniformly Continuous Map Between Metric Spaces, formed with $d_{X}$ and the distance $d_{\mathbb{R}}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv). Pointwise convergence and uniform convergence on a subset of $A$ are those of Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps, read with $D=E=A$.

(i) 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 $S$.

(ii) Let $x\in A$, let $r$ be a real number with $r>0$, and let $S\subseteq A$ contain 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 $A$.

(iii) 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 $A$.

(iv) 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 $A$.