Continuity for Sobolev mappings with null Lagrangian bounds

We prove the continuity of Sobolev functions $\phi \in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega })$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$, that satisfy $$\mid \mathrm{\nabla }\phi (x){\mid }^n\le K(x)(⟨\mathrm{\nabla }\phi (x),\xi (x)⟩+A(x)),$$ where $\xi \in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{n\mathrm{/}(n-1)}(\mathrm{\Omega },{\mathbb{R}}^n)$ is weakly divergence-free, and $K\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$, $A\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^q(\mathrm{\Omega })$ are non-negative with $p^{-1}+q^{-1}<1$. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$ satisfying the distortion inequality with defect $\mid Df(x){\mid }^n\le K(x)\det Df(x)+\mathrm{\Sigma }(x)$; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets.