An extendability condition for bilipschitz functions

We give a new definition of $\lambda$-relatively connected set, some generalization of a uniformly perfect set. This definition is equivalent to the old definition for large $\lambda$ but makes it possible to obtain stable properties for small $\lambda$. We prove the $\lambda$-relative connectedness of Cantor sets for corresponding $\lambda$. The main result is as follows: $A\subset\mathbb R$ admits the extension of all $M$-bilipschitz functions $f\colon A\to\mathbb R$ to $M$-bilipschitz functions $F\colon\mathbb R\to\mathbb R$ if and only if $A$ is $\lambda$-relatively connected. We give exact estimates of the dependence of $M$ and $\lambda$.