TITLE: Is the image of a null set under a differentiable map always null?
QUESTION [20 upvotes]: Let $n$ be a positive integer.  $\;  $  Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be everywhere Frechet differentiable.

Let $S$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero.
Does it follow that $\{f(s) : s\in S\}$ has Lebesgue measure zero?
(I know it would if $D(f)$ is continuous.)

REPLY [2 votes]: More generally, if $f:\ X \to Y$ is a locally Lipschitz function from one $\sigma$-compact metric space to another and $S \subseteq X$ has  $d$-dimensional Hausdorff measure 0, then $f(S)$ also has $d$-dimensional Hausdorff measure 0.  This is pretty much immediate from the definition of Hausdorff measures.  In your case, Lebesgue measure on ${\mathbb R}^n$ coincides with $n$-dimensional Hausdorff measure.