= Jump-amplitude inequality for total variation denoising
{title2=$[u]_{\nu_u}([f]_{\nu_u}-[u]_{\nu_u})\ge0\quad\mathcal H^{n-1}\text{-a.e. on }J_u$}
For scalar <total variation denoising> with $f\in BV(\Omega)\cap L^2(\Omega)$ on a bounded <Lipschitz domain> in arbitrary dimension, the inequality holds with common oriented <BV traces on a hypersurface>. It says that every surviving jump has the same direction as its data jump and no larger amplitude. Reversing the normal changes both differences and preserves the inequality.
Use <residual-preserving clipping of an ROF minimizer> and the <jump-amplitude inequality for a bounded ROF minimizer>. At each finite jump, choose an integer clipping level above both output traces. The clipped function has those same traces, and its residual-adjusted data have the same traces as $f$. Taking a countable union over the integer clipping levels proves the result. This gives the <no-new-jumps property of total variation denoising> under the full $L^2\cap BV$ assumptions, without asserting regularity of unbounded-forcing perimeter <minimizers>.
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