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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, n~ 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--~ R n, n~ 2, close to a boundary point xo E~ G assuming that f omits a set with positive capacity and that 8G is small at x. As local smallness measuresof BG we employ the following o two conditions• The first one is the continuum criterium which was recently introduced by Martio~ I]. See also [4]. The second one is the dispersion condition which was introduced by the author in 01" It was shown in [5] that the continuum criterium implies the dispersion condition but not conversely.