1. Databases
2. Proof database
3. Polynomials are smooth

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

1. Polynomials of any degree are continuous
2. Derivatives of a polynomial are also polynomials

Polynomials are continuous