Stability in the $C$-norm and $W^1_\infty$ of classes of Lipschitz functions of one variable

In the framework of Kopylov's $\omega$-stability concept, we study some stable classes of Lipschitz functions of one real variable. We obtain an exhaustive (nontrivial) classification for these classes and establish the relevant stability estimates in the $W^1_\infty$-norm.