Continuous one-to-one mapping from a subset $K \subset \mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Question

Let $f\colon \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a continuous function and $K\subset \mathbb{R}^n$ a subset of positive Lebesgue measure. Is it possible that $f$ is one-to-one on $K$?

If $K$ contains a (nonempty) open set this is impossible because of the invariance of domain theorem. But can we say anything for arbitrary (measurable) sets?

Answer 1

Let $n = 2$, $C \subseteq \mathbf R$ a fat cantor set, $f \colon C \times C \to C$ a homeomorphism. As $C \times C \subseteq \mathbf R^2$ is closed, there is a continuous extension $F \colon \mathbf R^2 \to [0,1]$ by the Tietze extension theorem. Now $\lambda(C \times C) > 0$ and $F|_{C \times C} = f$ is one-to-one.