Villani, Alfonso On Lusin’s condition for the inverse function. (English) Zbl 0562.26002 Rend. Circ. Mat. Palermo, II. Ser. 33, 331-335 (1984). The following theorem is proved. Let \(U\subset R^ n\) be open and \(\phi:U\to R^ n\) be continuous and one-to-one. If \(\phi\) is differentiable almost everywhere on U, then the inverse \(\phi^{-1}\) satisfies the Lusin’s condition (N) if and only if the Jacobian \(J_{\phi}\neq 0\) almost everywhere on U. Some known corollaries for the functions \(f:<a,b>\to R\) are added. Reviewer: A.Neubrunnová Cited in 15 Documents MSC: 26B05 Continuity and differentiation questions 26B10 Implicit function theorems, Jacobians, transformations with several variables Keywords:differentiability almost everywhere; inverse; Lusin’s condition (N); Jacobian × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Chiarenza F., Frasca M.,Boundedness for the Solutions of a Degenerate Parabolic Equation, to appear on Applicable Anal. · Zbl 0522.35059 [2] Hewitt E., Stromberg K.,Real and Abstract Analysis, Springer-Verlag, (1965). · Zbl 0137.03202 [3] McShane E. J.,Integration, Princeton University Press (1944). [4] Rudin W.,Real and Complex Analysis (2nd edition), McGraw-Hill (1974). · Zbl 0278.26001 [5] Rudin W.,Well-Distributed Measurable Sets, Amer. Math. Monthly,90 (1983), 41–42. · Zbl 0504.28003 · doi:10.2307/2975692 [6] Väisälä J.,Lectures on n-Dimensional Quasiconformal Mappings, Springer-Verlag (1971). · Zbl 0221.30031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.