I'm looking for a complete answer to this problem.
Let $U,V\subset\mathbb{R}^d$ be open sets and $\Phi:U\rightarrow V$ be a homeomorphism. Suppose $\Phi$ is differentiable in $x_0$ and that $\det D\Phi(x_0)=0$. Let $\{C_n\}$ be a sequence of open(or closed) cubes in $U$ such that $x_0$ is inside the cubes and with its sides going to $0$ when $n\rightarrow\infty$. Denoting the $d$-dimension volume of a set by $\operatorname{Vol}(.)$, show that $\lim_{n\rightarrow\infty}\frac{\operatorname{Vol}(\Phi(C_n))}{\operatorname{Vol}(C_n)}=0$
I know that $\Phi$ cant be a diffeomorphism in $x_0$, but a have know idea how to use this, or how to do anything different. Thanks for helping.