Recently I have read a paper *Weighted Trudinger-type Inequalities* written by Stephen M. Buckley and Julann O'Shea and published by Indiana University Mathematics Journal in 1999. I have some questions of pp.95-96 of this paper. 

Now let me state the background of this paper. Let $\Omega\subset \mathbb{R}^n\ (n\geqslant 2)$ be a domain (i.e., open connected set). Given a point $x_0\in \Omega$. Let $\{S_i\}_{i=0}^j,\ j\geqslant 2$ be a pairwise disjoint collection of open subsets of $\Omega$, such that 
\begin{equation*}
  \text{$x_0\in S_0$ and $x_0$ is in different connected components of $\Omega\setminus\overline{S_i}$ for all $0<i<j$.}
\end{equation*}
Define
\begin{equation*}
  l_i(y)=\inf_{\lambda\in \mathcal{F}_{y,x_0}}\mathrm{len} (\lambda\cap S_i),\quad y\in \Omega,\quad 0<i<j,
\end{equation*}
where $\mathcal{F}_{y,x_0}$ is the set of all rectifiable curves in $\Omega$ joining $y$ and $x_0$, and $\mathrm{len}(\lambda\cap S_i)$ stands for the arc length of $\lambda$ lying in $S_i$. Then $l_i$ is Lipschitz and
\begin{equation}\tag{*}\label{*}
  |\nabla l_i|\leqslant 1.
\end{equation}

**My questions are:**

1. Why $l_i$ is Lipschitz?

2. Why \eqref{*} holds?

For those questions, I am sure that $l_i$ is locally Lipschitz with constant $1$ on $\Omega$, that is, any point in $\Omega$ has a neighborhood on which $l_i$ is Lipschitz with constant $1$. Hence by Rademacher's theorem, we know that $l_i$ is differentiable a.e. in $\Omega$. But question 1 and 2 seem require $l_i$ is (global) Lipschitz with constant $1$ on $\Omega$, that is, for any $x',x''\in \Omega$, 
\begin{equation}\tag{**}\label{**}
  |l_i(x')-l_i(x'')|\leqslant |x'-x''|.
\end{equation}
I do not know how to prove the \eqref{**}.