Continuity Between Metric Spaces is Equivalent to Sequential Continuity

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces, let $A\subseteq X$, let $f:A\to Y$ and let $x\in A$.

(i) If $f$ is continuous at $x$ relative to $A$, then for every sequence $(y_{n})$ in $X$ with $y_{n}\in A$ for every $n\in\mathbb{N}$ that converges to $x$ in $(X,d_{X})$, the sequence $(f(y_{n}))$ converges to $f(x)$ in $(Y,d_{Y})$.

(ii) Assume the axiom of dependent choice. If for every sequence $(y_{n})$ in $X$ with $y_{n}\in A$ for every $n\in\mathbb{N}$ that converges to $x$ in $(X,d_{X})$ the sequence $(f(y_{n}))$ converges to $f(x)$ in $(Y,d_{Y})$, then $f$ is continuous at $x$ relative to $A$.

(iii) Assume the axiom of dependent choice. Then $f$ is continuous on $A$ if and only if for every $x'\in A$ and every sequence $(y_{n})$ in $X$ with $y_{n}\in A$ for every $n\in\mathbb{N}$ that converges to $x'$ in $(X,d_{X})$, the sequence $(f(y_{n}))$ converges to $f(x')$ in $(Y,d_{Y})$.