Continuity of a Real Function Agrees with Metric Continuity on the Real Line

Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $E\subseteq\mathbb{R}$, let $f:E\to\mathbb{R}$, and let $x_0\in E$.

Then the following two statements are equivalent.

1. $f$ is continuous at $x_0$.

2. $f$ is continuous at $x_0$ relative to $E$, as a map from the subset $E$ of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$.

Consequently, $f$ is continuous at every point of $E$ in the sense of statement 1 if and only if $f$ is continuous on $E$ in the sense of statement 2.