There's some ambiguity around these terms so I'll be as clear as I can.
A set $A \subseteq R$ is said to be negligible if for all $\epsilon > 0$ there are boxes $Q_1, Q_2, \dots$ such that $A \subset \bigcup Q_i$ and $\sum V(Q_i) < \epsilon$
A set $E \subseteq \mathbb R$ is said to be Jordan measurable if $\partial E$ is negligible.
The proposition I wish to prove is that if $f: \mathbb R \to \mathbb R$ is $C^1$ and $E \subset \mathbb R$ is Jordan measurable then $f(E)$ is also Jordan measurable.
What I tried:
Not sure if this is the correct direction. Let $\epsilon > 0$. $E$ is Jordan measurable so there are closed intervals $I_1, I_2, \dots$ such that $\partial E \subset \bigcup I_k$ and $\sum V(I_k) < \epsilon$.
$f$ is $C^1$ on $I_k$ and so it's Lipschitz there. We know that Lipschitz functions map negligible sets to negligible sets, and so $f(I_k)$ is negligible for all $k$.
It is clear to see now that $f(\partial E) \subset\bigcup f(I_k)$ is negligible, since it's a countable union of negligible sets.
But we weren't asked about $f(\partial E)$. We were asked about $\partial f(E)$. Is there any relation between the two? If $f$ is a diffeomorphism, then I know that they are equal. But can we say anything when $f$ is not know to be invertible, but rather just $C^1$?