How to calculate the Lipschitz constant(or Lipschitz norm) for $|x|$? I want to calculate the following equality:
$$
\sup_{x\in[-a,a]} |x| + \sup_{x,x' \in [-a,a]} \frac{||x'|-|x||}{|x'-x|} = a+1
$$
for $a>0$.
The first term supremum is $a$, but I cannot know how to calculate the second term.

Hint: first show that the second supremum is less than or equal to one. This follows from the triangle inequality.
Then fix $\varepsilon>0$ and try to pick two numbers $x$ and $x'$ in $[-a,a]$ such that
$$
\frac{||x'|-|x||}{|x'-x|} > 1-\varepsilon.
$$
It will be rather easy if you look for these numbers in $[0,a]$ or $[-a,0]$. Actually, you can even do better than this.
