proving a  function satisfies a  Lipschitz condition 
Let $F$ be a closed set of $ \mathbb{R} $ whose complement has finite measure. Let $\delta(x) = d (x, F) =\inf \{ |x - z| \mid z \in F\}$.
Prove $ \delta$ continuous by proving $| \delta(x) - \delta(y) | \leq |x - y|$

I appreciate any kind of hint.
Thanks

I don't think we need $F$ to be a closed set whose complement has finite measure etc. The result holds for any nonempty $F$.
Hint: $|x-z|\le |x-y|+|y-z|$ and take $\inf$ over $z\in F$.
