The Absolute Value on the Real Line is 1-Lipschitz and Continuous

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $d_{\mathbb{R}}$ be the distance on $\mathbb{R}$, and let $\alpha:\mathbb{R}\to\mathbb{R}$ be the map $\alpha(x)=|x|$, with the absolute value of $x$.

(i) Lipschitz bound. $\alpha$ is Lipschitz with constant $1$ as a map from $(\mathbb{R},d_{\mathbb{R}})$ to $(\mathbb{R},d_{\mathbb{R}})$, that is,

$$\bigl||x|-|y|\bigr|\le|x-y|\qquad(x,y\in\mathbb{R}).$$

(ii) Continuity. $\alpha$ is continuous on $\mathbb{R}$.