Problem:

Suppose that $\displaystyle p:R \rightarrow R$ is a polynomial. Show that $\displaystyle p:R \rightarrow R$is Lipschitz if and only if the degree of the polynomial is less than 2.

Attempt:

By definition of Lipschitz, then we need to prove that the functionpis Lipschitz for degree less than 2 or equivalently $\displaystyle \leq 1 $ then

$\displaystyle | p(x_{1}) - p(x_{2}) | \leq K |x_{1} - x{2}| $ for $\displaystyle K \geq 0 $ and for all $\displaystyle x, x_{1} \in R $

Here I'm not sure how to approach this. Since p is a polynomial, then it must be continuous. I was thinking of showing that ifpis a polynomial of less than 2, then its derivative is a constant. Thus, its derivative is bounded and therefore a Lipschitz function.

Thank you for your time.