Let U ∋ 0 be a hyperbolic domain, α ∈ R\Q, let ∆ be a Stolz angle at λ0 = e2πα with respect to the unit disk D, and W a domain containing the point λ0. Consider an analytic family f ⋅ W × U → C ; (λ, z) ↦ fλ(z) consisting of analytic functions in the domain U with the following expansion fλ(z) = λz + α2(λ)z2 + , λ ∈ W, for small z. Let A* (0, fλ, U) be the maximal domain A ⊂ U, such that 0 ∈ A and fl (A) ⊂ A, or the set {0} if there exist no such domains. We prove, that if a sequence {λ0 ∈ W ∩ Δ} n∈N converges to λ0 and S := A* (0, fλ, U) ≠ {0},  then the sequence of the domains A* (0, fλ, U) converges to S as to the kernel. An example shows, that the analogous statement for convergence with respect to the Hausdorff metric does not hold. In the case S ⊂ U we obtain an asymptotic estimate for the size of the neighbourhood V = V (K) of the point λ0 , such that a given compact K ⊂ S lies in A* (0, fl, U) for all λ ∈ V ∩ Δ.