= Scalar total variation splitting under clipping
{title2=$\operatorname{TV}(u)=\operatorname{TV}(T_Mu)+\operatorname{TV}(u-T_Mu)$}
For a scalar <BV space> function, clipping to $[-M,M]$ partitions its <coarea formula for BV functions> into middle and tail levels. The clipped function has the original <superlevel sets> at heights in $(-M,M)$. The positive tail levels of $u-T_Mu$ correspond to original heights above $M$, and its negative tail levels to heights below $-M$. Thus their <total variation seminorms> add exactly. This is stronger than the <triangle inequality> and is specific to scalar monotone clipping. It supports <residual-preserving clipping of an ROF minimizer>.
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