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

Distances. The distance $d_{\mathbb{R}}$ is the map $\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ with values $|s-t|$, which exists and is unique by The Distance on the Real Numbers Is a Map with Values the Absolute Differences (i); so for $s,t\in\mathbb{R}$, $d_{\mathbb{R}}(s,t)=|s-t|$ by The Distance on the Real Numbers (i), and $|s-t|=|-(t-s)|=|t-s|$, as $s-t=-(t-s)$ (Difference and Quotient Identities in ℝ (i)) and $|-w|=|w|$ for $w\in\mathbb{R}$ (Rules of Arithmetic and Order in an Ordered Field (x)). Let $x'\in E$ and $y\in E$. As $E\subseteq\mathbb{R}$, $x',y\in\mathbb{R}$, and $f(y),f(x')\in\mathbb{R}$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (ii)). Hence

$$d_{\mathbb{R}}(x',y)=|y-x'|\qquad\text{and}\qquad d_{\mathbb{R}}(f(y),f(x'))=|f(y)-f(x')| .$$

Proof of (i). By Continuous Map Between Metric Spaces (i), with $(X,d_{X})=(Y,d_{Y})=(\mathbb{R},d_{\mathbb{R}})$ and $A=E$ (a set, as a subclass of the set $\mathbb{R}$, by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (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 $d_{\mathbb{R}}(x,y)<\delta$ satisfies $d_{\mathbb{R}}(f(y),f(x))<\varepsilon$. By the two equalities above, with $x'=x$, this condition is the one stated.

Proof of (ii). By Continuous Map Between Metric Spaces (ii), $f$ is continuous on $E$ if and only if it is continuous at $x'$ relative to $E$ for every $x'\in E$. By (i), applied at each $x'\in E$ in place of $x$, this is the stated condition.