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In the paper we investigate continuity of Orlicz-Sobolev mappings$W^{1,P}(M,N)$ of finite distortion between smooth Riemannian $n$-manifolds,$n\geq 2$, under the assumption that the Young function $P$ satisfies the socalled divergence condition $\int_1^\infty P(t)/t^{n+1}\, dt=\infty$. We provethat 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. Thatincludes mappings with $Df\in L^n$ and, more generally, mappings with $Df\inL^n\log^{-1}L$. On the other hand, if the space $W^{1,P}$ is larger than$W^{1,n}$ (for example if $Df\in L^n\log^{-1}L$), and the universal cover of$N$ is homeomorphic to $\mathbb{S}^n$, $n\neq 4$, or is diffeomorphic to$\mathbb{S}^n$, $n=4$, then we construct an example of a mapping in$W^{1,P}(M,N)$ that has finite distortion and is discontinuous. Thisdemonstrates a new global-to-local phenomenon: both finite distortion andcontinuity are local properties, but a seemingly local fact that finitedistortion implies continuity is a consequence of a global topological propertyof the target manifold $N$.

Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion