TITLE: A curve whose image has positive measure
QUESTION [9 upvotes]: It is well-known that there are continuous curves $f:I \to \mathbb R^2$ (where $I \subset \mathbb R$ is an interval) whose image have positive measure (e.g Peano curve).
I have read somewhere that if we require the curve to be differentiable evrywhere then this cannot happen; but if we require it to be almost everywhere differentiable, then it can happen! How could one proceed to prove the first statement and give a counterexample for the second?

REPLY [2 votes]: The other direction is called (mini)-Sard's Theorem. Page 205 in Appendix 1 of Guilleman and Pollack, Differential Topology. The mini version is just this: Let $U$ be an open set of $\mathbb R^n,$ and let $f:U \rightarrow \mathbb R^m$ be a smooth map. Then, if $m > n,$ we can conclude that $f(U)$ has measure zero in $\mathbb R^m.$
The full Sard's Theorem is about critical points, while we make no demands about the relative dimensions. Let $f:X \rightarrow Y$ be a smooth map of manifolds, and let $C$ be the set of critical points of $f$ in $X.$ Then $f(C)$ has measure zero in $Y.$ 
They say their proof is almost verbatim from John W. Milnor, Topology from the Differentiable Viewpoint. It appears Guillemin and Pollack, also Milnor, assume $C^\infty.$ However, Milnor refers back to Pontryagin(1955), English translation (1959). Evidently  Pontryagin worked with weaker conditions, but we are not told what. http://en.wikipedia.org/wiki/Sard%27s_theorem
Alright, I've been looking stuff up elsewhere. The full Sard's theorem does depend on relative dimension. I currently think that mini-Sard make be true both for $C^1$ and for differentiable everywhere, which is a pretty weak hypothesis.