Mappings of generalized finite distortion and continuity

We study continuity properties of Sobolev mappings (Formula presented.), (Formula presented.), that satisfy the following generalized finite distortion inequality (Formula presented.) for almost every (Formula presented.). Here (Formula presented.) and (Formula presented.) are measurable functions. Note that when (Formula presented.), we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion (Formula presented.), where a sharp condition for continuity is that (Formula presented.) is in the Zygmund space (Formula presented.) for some (Formula presented.). We also show that one can slightly relax the boundedness assumption on (Formula presented.) to an exponential class (Formula presented.) with (Formula presented.), and still obtain continuous solutions when (Formula presented.) with (Formula presented.). On the other hand, for all (Formula presented.) with (Formula presented.), we construct a discontinuous solution with (Formula presented.) and (Formula presented.), including an example with (Formula presented.) and (Formula presented.).