Continuity of the Identity Map, of Powers, and of Polynomial Functions on a Subset of the Real Line

Let $\mathbb{R}$ be the real numbers, with the addition, multiplication and order of their ordered field structure, let $|\cdot|$ be the absolute value, and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ with the metric $d_{\mathbb{R}}(s,t)=|s-t|$. Let $\mathbb{N}$ be the set of natural numbers with successor map $S$, and let $x^{n}$ denote the $n$th power of $x$.

Let $E\subseteq\mathbb{R}$. Throughout, a map $E\to\mathbb{R}$ is called continuous on $E$ if it is continuous on $E$ as a map from the subset $E$ of the real line into the real line.

Then the following hold.

1. (Identity) ¶ The map $E\to\mathbb{R}$ given by $x\mapsto x$ is continuous on $E$.

2. (Powers) ¶ For every $n\in\mathbb{N}$ the map $E\to\mathbb{R}$ given by $x\mapsto x^{n}$ is continuous on $E$.

3. (Polynomial functions) ¶ For every polynomial function $p$ on $\mathbb{R}$, the restriction $p|_{E}$ is continuous on $E$.