Let
$sat_{\bar{u}}(u) = sign(u)\min({|u|, \bar{u}}), \bar{u}>0$, be a saclar function from $\Bbb{R}$ to $[-\bar{u}, \bar{u} ]$.
$dz_{\bar{u}}(u) = sign(u)\max({0, |u|-\bar{u}}), \bar{u}>0$, be a saclar function from $\Bbb{R}$ to $\Bbb{R}$.
I want to prove that $sat_{\bar{u}}$ and $dz_{\bar{u}}$ are globally 1-Lipscitz functions.
Can anyone please help me out here?
Thanks a lot