Geodesic
Continuity in $\Lambda$-variations of functions of several variables
Sbornik. Mathematics, Tome 194 (2003) no. 7, pp. 1009-1034

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A test for the coincidence in the multidimensional case of the class of functions of bounded $\Lambda$-variation with the class of functions continuous in $\Lambda$-variation is obtained for a large variety of sequences $\Lambda$ (such that the ratio $\lambda_{2n}/\lambda_n$ has a limit as $n\to\infty$). 
@article{SM_2003_194_7_a3,
     author = {O. S. Dragoshanskii},
     title = {Continuity in  $\Lambda$-variations of functions of several variables},
     journal = {Sbornik. Mathematics},
     pages = {1009--1034},
     publisher = {mathdoc},
     volume = {194},
     number = {7},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_7_a3/}
}
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O. S. Dragoshanskii. Continuity in  $\Lambda$-variations of functions of several variables. Sbornik. Mathematics, Tome 194 (2003) no. 7, pp. 1009-1034. http://geodesic.mathdoc.fr/item/SM_2003_194_7_a3/