I'm studying example 4.1.3 of "Variational Analysis in Sobolev Spaces and BV Spaces" by Attouch, Buttazzo and Michaille. At one point it states that
Given $f: R^n \to R^m \> (n\leq m)$ one to one of class $C^1(R^n)$ (i.e. with all partial derivatives continous) and given a compact $E$ of $R^n$ Then $n$-dimensional Hausdorff measure of f(E) is greater than zero.
My approach:
Being $E$ compact and $f\in C^1(R^n)$ follows $f$ is Lipschtiz on $E$, so from proposition 4.1.6 of the same book one has $H^n(f(E))=\int_E \Big( \sum_{i=1}^C |J_i|^2 \Big)^{\frac{1}{2}} d\>m_n$ where $J_i$ are the minors $n\times n$ of Jacobian matrix $J(f)$
By this way is enough to prove that not all minors $n\times n$ are equals to zero that is the rank of Jacobian is max that is the differential of $f$ is injective. If f has been $R^n \to R^n$ it would be simple from composition of $J(f)$ $J(f^{-1})$ but in this case I don't know the right route.