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The study of quasiregular mappings during the last decade has shown that these mappings are natural generalizations of analytic functions of complex plane to n—dimensional euclidean spaces, na 2.'The basic results on these mappings can be found in [2] and [3]. In this paper we study the behavior of a quasiregular mapping f: G—~» RP, n 2-2, close to a boundary point x0 6 36 assuming that f omits a set with positive capacity and that 6G is small at ‘xo. As local smallness measuresof 3G we employ the following two conditions. The first one is the continuum criterium which was recently introduced by Martio [1]. See also [4]. The second one is the dispersion condition which was introduced by the author 1115}. It was shown in [5] that the continuum criterium implies the dispersion condition but not conversely. We shall prove following results. If f: G—u—Rn is a quasiregular mapping omitting a set with positive …