Bounded-variation contraction under clipping

For $T(t)=\min (b,\max (a,t))$ and $u\in BV(\Omega )$, $T\circ u\in BV(\Omega )$ and $|D(T\circ u)|(\Omega )\le |Du|(\Omega )$. This is the contraction property of a one-Lipschitz scalar composition, obtainable by smooth approximation and sequential lower semicontinuity. It proves range preservation for total variation denoising when the data lie in $[a,b]$.