# The improvement of Sard's Theorem

If $f:\mathbb{R}\to \mathbb{R}$ is an absolutely continuous function and let $C$ be the set of all $x\in\mathbb{R}$ such that $f'(x)=0$, is it necessarily that $f(C)$ is a Lebesgue $0$ set?

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What does $f'$ mean in this level of generality? – David Speyer Mar 20 '12 at 16:04
Since $f$ is absolutely continuous, its derivative exists almost everywhere. – Hezudao Mar 24 '12 at 3:11

The set $\{f(x):f'(x)=0\}$ is Lebesgue null for any function $f:\mathbb R\to\mathbb R$ whatsoever; it does not have to be absolutely continuous, or even continuous, or even measurable. (If the derivative does not exist at $x$, the condition $f'(x)=0$ is not fulfilled.)
Proof. It suffices to prove that $\{f(x): x\in [0,1], \ f'(x)=0\}$ is null. Fix $\epsilon>0$. Cover the set $A=\{x\in [0,1], \ f'(x)=0\}$ by all intervals of the form $I(x,\delta)=(x-\delta,x+\delta)$ where $0<\delta<1$ and $\operatorname{diam} f(I(x,5\delta))\le \epsilon \,\delta$. Using Vitali covering lemma, choose a disjoint set of intervals $I(x_j,\delta_j)$ such that the intervals $I(x_j,5\delta_j)$ still cover $A$. Disjointness implies $\sum 2\delta_j\le 3$. The sets $f(I(x_j,5\delta_j))$ cover $f(A)$ and have $$\sum \operatorname{diam} f(I(x_j,5\delta_j)) \le \sum \epsilon\delta_j\le \frac32 \epsilon$$ Since $\epsilon>0$ was arbitrary, $f(A)$ is Lebesgue null. $\Box$