Relation between Lipschitz function and $W^{1,\infty}$

Question

In a bounded Star shaped domain $\Omega$;Let A function to be $f$ on $\Omega$ I want to fin the relation between:

1)$f$ is Lipschitz in $\Omega$.

2)$f$ is in $W^{1,\infty}$.

If we have 1) did we have that: $\exists M>0$ such that: $\left | \frac{\partial f}{xi} \right |<M$ a.e ?

Answer 1

Yes, if you have $f$ is Lipschitz in $\Omega \subset \mathbb{R}$, so $|f'| \leq M$, for some $M$.