# Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null.

I think it has something to do with the fact that $f'$ is bounded in any interval, but I'm at a complete loss as to how to take it further. Any help much appreciated.

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What you might try is to prove it for an interval first, and then use the fact that if $A$ is null, then for any $\epsilon > 0$, there exist intervals $(I_n)_{n\in\mathbb{N}}$, such that $\sum_{n=1}^\infty |I_n| < \epsilon$, and $A \subset \cup_{n\in\mathbb{N}} I_n$. To prove it if $A$ is a single interval, you do indeed need to use the fact that $f'$ is bounded in any compact interval (i.e. $\overline{I_n}$) –  Nicholas Stull Apr 26 '12 at 1:51
Do you know about absolutely continuous functions? They preserve null sets, and any function with a continuous derivative is absolutely continuous. –  Patrick Apr 26 '12 at 5:25