Values of Finite Distortion: Continuity

Abstract: We prove continuity for mappings $f \colon \Omega \to \mathbb{R}^n$, $\Omega \subset \mathbb{R}^n$, in the Sobolev class $W^{1,n}_{\text{loc}}(\Omega, \mathbb{R}^n)$ that satisfy the inequality [ \lvert Df(x) \rvert^n \le K(x) \det Df(x) + \Sigma(x) ] whenever $K \in L^p_{\text{loc}}(\Omega)$ and $\Sigma \in L^q_{\text{loc}}(\Omega)$ with $p^{-1} + q^{-1} < 1$. This closes a significant gap between existing methods and known counterexamples. The result is sharp, new even in the planar case, and opens a systematic study of mappings with values of finite distortion in geometric function theory. As a key part of the proof, we introduce an overlooked Sobolev-type inequality based on measures of superlevel sets.