Does global lipschitz imply locally lipschitz?

Question

This is just a quick question:

I'm a little confused as to whether or not globally Lipschitz continuous implies Locally Lipschitz?

I'm aware that if $f$ is globally lipschitz, it means there is a positve $K$ such that for all $x, y \in \mathbb{R}^n$ then:

\begin{equation} |f(x)-f(y)|\leq K|x-y| \end{equation}

If it's locally, then for every point $a$ contained in an open subset of $\mathbb{R}^n$, there exists a small neighborhood around $a$ and a positive constant $L$ such that for all $x,y \in N_{\delta}(a)$ then \begin{equation} |f(x)-f(y)|\leq L|x-y| \end{equation}

I would think it global implies local, but I'm not sure to be honest.

Thank you for your time.

Answer 1

Yes, it does. Take $L=K$.

Dot dot dot.