We prove that an open discrete Q-mapping \( f:D \to \overline {{\mathbb{R}^n}} \) has a continuous extension to an isolated boundary point if the function Q(x) has finite mean oscillation or logarithmic singularities of order at most n – 1 at this point. Moreover, the extended mapping is open and discrete and is a Q-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on Q-mappings. In particular, we prove that an open discrete Q-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 116–126, January, 2009.
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Sevost’yanov, E.A. Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for Q-mappings. Ukr Math J 61, 140–153 (2009). https://doi.org/10.1007/s11253-009-0190-4
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DOI: https://doi.org/10.1007/s11253-009-0190-4