Is the Lipschitz constant the maximum norm of the gragient?

Question

1. Is the Lipschitz constant $L$ of $f$ just the $max(\|\nabla f\|_2)$, or there are some unobvious differences?
2. If the function $f$ is Lipschitz only on set $S$, is it correct to define Lipschitz constant $L$ for cases like that?

Answer 1

Ad 2: Yes.

Ad 1: When the considered domain $S\subset{\mathbb R}^n$ is convex then $\max_{x\in S}\bigl(\|\nabla f(x)\|_2\bigr)$ is an acceptable Lipschitz constant for $f$. But otherwise things may happen you don't like. Consider the slit domain $$S:=\bigl\{(x,y)\in{\mathbb R}^2\bigm| x>0 \ {\rm or}\ |y|>0, \ x^2+y^2\geq1\bigr\}\ .$$ The principal value ${\rm Arg}(x,y)\in\ ]-\pi,\pi[\ $ of the polar angle $\phi(x,y)$ is a nice function on $S$, but is not Lipschitz continuous on $S$. (Consider two points $(-R,\pm\epsilon)\in S$ with $R\gg1$, $\>\epsilon\ll1$.)