Commutator Lipschitz functions and analytic continuation

Let $\mathfrak F_0$ and $\mathfrak F$ be perfect subsets of the complex plane $\mathbb C$. Assume that $\mathfrak{F_0\subset F}$ and the set $\Omega\stackrel{\mathrm{def}}=\mathfrak{F\setminus F}_0$ is open. We say that a continuous function $f\colon\mathfrak F\to\mathbb C$ is an analytic continuation of the function $f_0\colon\mathfrak F_0\to\mathbb C$ if $f$ is analytic on $\Omega$ and $f|\mathfrak F_0=f_0$. In the paper it is proved that if $\mathfrak F$ is bounded, then the commutator Lipschitz seminorm of the analytic continuation $f$ coincides with the commutator Lipschitz seminorm of $f_0$. The same is true for unbounded $\mathfrak F$ if some natural restrictions concerning the behavior of $f$ at infinity are imposed.