Show $\lambda^{n}(G(A))=0$ where $A:= \{x \in \mathbb R^{n} : dG_{x} = 0\}$

Question

Let $G : \mathbb R^{n} \to \mathbb R^{n}$ be a $C^{1}$-map and $A:= \{x \in \mathbb R^{n} : dG_{x} = 0\}$.

Show that $\lambda^{n}(G(A))=0$

What I know: I have previously shown that for a Lipschitz function $F$ that $\lambda^{n}(F(M))\leq C L^{n}\lambda^{n}(M)$

And since $G$ is $C^{1}$ it follows that we know that $\vert \vert G(x) -G(y)\vert \vert \leq M \vert \vert x-y\vert \vert$, hence Lipschitz. But how can this help me?

Answer 1

This a straight forward application of Sard's theorem. It states that the set of points in which the derivative is not an isomorphism has measure zero in the codomain. In this case $A$ is contained in such set.