$W^{s,\frac{n}{s}}$-maps with positive distributional Jacobians

We extend the well-known result that any $f\in W^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(\mathrm{\Omega })$ for any $s\ge \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac{n}{s}}$ of the degree estimates known for $W^{1,n}$-maps with positive or non-negative Jacobian, such as the sense-preserving property.