  Let $\Omega\subset\mathbb{R}^3$ be a domain and let $f\in
BV_{\operatorname{loc}}(\Omega,\mathbb{R}^3)$ be a homeomorphism such that its
distributional adjugate is a finite Radon measure. We show that its inverse has
bounded variation $f^{-1}\in BV_{\operatorname{loc}}$. The condition that the
distributional adjugate is finite measure is not only sufficient but also
necessary for the weak regularity of the inverse.
