How to calculate the Lipschitz constant(or Lipschitz norm) for $|x|$?

Question

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.

Answer 1

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.

Answer 2

We can try as follows: suppose $\;x'\ge x\;$ , then

$$x\ge0:\;\;\implies \begin{cases}|x'|-|x|=x'-x\\{}\\|x'-x|=x'-x\end{cases}\implies\;\frac{||x'|-|x||}{|x'-x|}=1$$

and now, in general, and for $\;x\neq x'\;$:

$$r:=\frac{||x'|-|x||}{|x'-x|}\implies r^2=\frac{x'^2-2|xx'|+x^2}{x'^2-2xx'+x^2}\le1$$

so you are done.