Continuity of Real Functions on Subsets of ℝ in ε-δ Form

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $d_{\mathbb{R}}$ be the distance on $\mathbb{R}$, a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Numbers (i), and read every map from a subset of $\mathbb{R}$ into $\mathbb{R}$ as a map from that subset of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$. Let $E\subseteq\mathbb{R}$, let $f:E\to\mathbb{R}$ and let $x\in E$.

(i) $f$ is continuous at $x$ relative to $E$ if and only if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $y\in E$ with $|y-x|<\delta$ satisfies $|f(y)-f(x)|<\varepsilon$.

(ii) $f$ is continuous on $E$ if and only if for every $x'\in E$ and every real $\varepsilon>0$ there is a real $\delta>0$ such that every $y\in E$ with $|y-x'|<\delta$ satisfies $|f(y)-f(x')|<\varepsilon$.