Polynomial Functions on the Real Line are Smooth

Let $\mathbb{R}$ be the real numbers, regarded as the Euclidean space $\mathbb{R}^{1}$, and let $p:\mathbb{R}\to\mathbb{R}$ be a polynomial function on $\mathbb{R}$.

Then the following hold.

1. (The whole space is open) $\mathbb{R}^{1}$ is an open subset of $\mathbb{R}^{1}$.

2. (Smoothness) For every open subset $U$ of $\mathbb{R}^{1}$ the restriction of $p$ to $U$ is smooth on $U$. In particular $p$ is smooth on $\mathbb{R}^{1}$, and hence of class $C^{k}$ on $\mathbb{R}^{1}$ for every natural number $k$.