# Polynomials are smooth

A polynomial of degree $n$ is defined as

\begin{equation} P[x] = \sum_{i = 0}^n a_i x^n \end{equation}for $a_i \in \mathbb{R}$ (the proof would be essentially the same for say $\mathbb{C}$). We need simply to show that

- Polynomials of any degree are continuous
- Derivatives of a polynomial are also polynomials