Bounded Continuous weak quasiregular mappings that fail to be quasiregular

We show that, in dimensions $n\geq 3$, continuity and boundedness do not restore the Sobolev regularity conjecture of Iwaniec and Martin for weakly quasiregular mappings below the critical exponent. For every bounded domain $\Omega\subset\mathbb R^n$ and every $1\leq p<nK/(K+1)$, we construct a bounded continuous weakly $K$-quasiregular mapping $$f\in W^{1,\,p}(\Omega;\,\mathbb R^n)\cap C(\Omega;\,\mathbb R^n) \cap L^\infty(\Omega;\mathbb R^n)$$ which fails to be quasiregular. We further construct weakly quasiregular mappings whose singular sets have Hausdorff dimension arbitrarily close to the maximal size permitted by their Sobolev regularity. These examples show that, the almost-everywhere sign condition on the Jacobian is too weak to serve as an orientation-preserving hypothesis below $W^{1,n}$. In contrast, we show that, for $n-1<p<n$, quasiregularity follows once this condition is replaced by a one-sided condition on the distributional degree (together with boundedness).