On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings $f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3}$ of a domain $D \subset {\mathbb{R}^3}$ satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point $b \in {\mathbb{R}^3}$, we prove the following statement: Let a point y ₀ belong to $\overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right)$ and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where $I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\}$ and $g:U \to {\mathbb{R}^n}$ is a quasiconformal mapping of a domain $U \subset {\mathbb{R}^n}$ such that A ⊂ U.