Geodesic
Criteria for the continuity of finite-dimensional representations
Sbornik. Mathematics, Tome 195 (2004) no. 9, pp. 1377-1391

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Necessary and sufficient continuity conditions for finite-dimensional (not necessarily topological) representations of connected locally compact groups are obtained. Namely, it is shown that a finite-dimensional representation of a connected locally compact group is continuous if and only if the oscillation of this representation at the identity element of the group is less than 2. 
@article{SM_2004_195_9_a7,
     author = {A. I. Shtern},
     title = {Criteria for the continuity of finite-dimensional representations},
     journal = {Sbornik. Mathematics},
     pages = {1377--1391},
     publisher = {mathdoc},
     volume = {195},
     number = {9},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2004_195_9_a7/}
}
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A. I. Shtern. Criteria for the continuity of finite-dimensional representations. Sbornik. Mathematics, Tome 195 (2004) no. 9, pp. 1377-1391. http://geodesic.mathdoc.fr/item/SM_2004_195_9_a7/