Towards a theory of removable singularities for maps with unbounded characteristic of quasi-conformity

We prove that sets of zero modulus with weight $Q$ (in particular, isolated singularities) are removable for discrete open $Q$-maps $f\colon D\to\overline{\mathbb R}{}^n$ if the function $Q(x)$ has finite mean oscillation or a logarithmic singularity of order not exceeding $n-1$ on the corresponding set. We obtain analogues of the well-known Sokhotskii–Weierstrass theorem and also of Picard's theorem. In particular, we show that in the neighbourhood of an essential singularity, every discrete open $Q$-map takes any value infinitely many times, except possibly for a set of values of zero capacity.