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We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\\in L^n$ and, more generally, mappings with $Df\\in L^n\\log^{-1}L$. 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We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\\in L^n$ and, more generally, mappings with $Df\\in L^n\\log^{-1}L$. 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