Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$ Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude that $f$ cannot be one-to-one on $K$?
Continuity is (in general) not enough, as demonstrated here.
On the other hand, a nonvanishing Jacobian on a subset of $K$ of positive measure allows us to construct a contradiction by the coarea formula. 
But what if we cannot assume anything about the Jacobian? Is, e.g., Lipschitz continuous sufficient to construct a contradiction? Or do there exist Lipschitz continuous examples of one-to-one mappings?
Edit: This seems to have a connection to singularity theory. Unfortunately, things like Sard's Theorem also don't help as it only tells me something about singular values but I would need some information about the possible size of singular points of a one-to-one mapping.

Kaufman [1] proved that there exists a surjection $f\in\mathcal{C}^1((0,1)^{n+1},(0,1)^n)$ such that $\operatorname{rank}(Df)\leq 1$ for all $x\in (0,1)^{n+1}$. Therefore I suppose you need less then $\mathcal{C}^1$, and probably less than Lipschtiz, but this is not trivial since Whitney is not a.e. 
Moreover, there is a related result for Sobolev functions by Wildrick and Zürcher [2].  Their Theorem 1.5 states that for any  $\epsilon > 0$, there is a compact Ahlfors $2$-regular metric space $X$ which supports a $(2 + \epsilon)$-Poincaré inequality, and is such that  for any $1 \le p < 2 +\epsilon$, there is a continuous surjection $f : X \to Y$ onto any length-compact metric space $Y$ that is constant off a set of finite measure and has an upper gradient in the space $L^p(X)$. 
References
[1] R. Kaufman. A singular map of a cube onto a square. J. Differential Geom. 14 (1979), no. 4, 593–594 (1981)
[2] K. Wildrick, T. Zürcher.  Space filling with metric measure spaces. 
Math. Z. 270 (2012), no. 1-2, 103–131. 
