On the local behaviour of the multidimensional $\Lambda$-variation

Let two classes $(\Lambda^1,\dots,\Lambda^m)BV$ and $(M^1,\dots,M^m)BV$ on an interval $\Delta$ be given. In the paper, we find necessary and sufficient conditions for the $\Lambda$-variation of any function in the $M$-class, over a neighbourhood of every regular point, to tend to zero as the neighbourhood decreases.
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