Locally Lipschitz and polynomial map

Question

I'm trying to prove that: if $p:\mathbb{R}^p \to \mathbb{R}^q$ is a polynomial map then $p$ is Lipschitz over compacts.

Answer 1

Polynomials are differentiable with continuous derivative, and their derivatives have a maximum operator norm somewhere on the compact set. This maximum norm should serve as a Lipschitz constant.