Let $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Do we also have that $$f = \int_{\mathbb R} \chi_{\{f >h\}} \, dh$$ holds?
2026-03-30 11:55:32.1774871732
Coarea-like formula for BV function (not its derivative)
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That depends on whether or not $$0 = \int_{-\infty}^0 1 \, dh.$$