 Solution

For $a,b\ge 0$ and $c\ge 1$,Apply this to the first component with $a=|z_2|$, $b=|w_2|$, $c=x^2+1$. The reverse triangle inequality givesFor the second component use $a=|z_1|$, $b=|w_1|$, $c=1$ to obtain $|F_2(x,Z)-F_2(x,W)|\le |z_1-w_1|$. Taking the maximum yieldsThus $F$ is globally Lipschitz continuous in the dependent vector, uniformly in $x$, with Lipschitz constant $1$. Differentiability of the absolute-value function at zero is unnecessary.