Does there exist a smooth function which may map a set of $0$ measure to a set of positive measure?
The devil's starircase+ $x$ is a function which maps the Cantor set to a set of positive measure, but it is not smooth (not even differentiable)
I am trying to prove Sard's Theorem, and the non-existence of such a function might make life easy for me in some special cases.
EDIT: Consider only the Lebesgue measure on both the domain and the range, which for now consider to be $\Bbb{R}$.
Best Answer
Assume $f\colon [0,1]\to\Bbb R$ is differentiable and $E\subset [0,1]$ is a zero set whereas $\mu(f(E))=1$. Then for any $\epsilon>0$, $E$ can be covered by intervals $A_i$ of total length $\epsilon$. Each $A_i$ is mapped to some interval $B_i$, and from $\sum|B_i|\ge1$, we see that $\frac{|B_i|}{|A_i|}>\frac1\epsilon$. By the MVT, $|f'(x)|>\frac1\epsilon$ for some $x\in A_i$. Thus $f'$ is not bounded on $[0,1]$ and cannot be continuous.