Why is Cantor step function not a counter example to Sard's Theorem

Question

I recently learned about Sard's theorem: Let $f:[a,b]\to\mathbb{R}$ and denote $E_f=\{x\in(a,b)|f'(x)\mbox{ exists and }f'(x)=0\}$, then $m(f(E_f))=0$. Now, Cantor's step function is almost everywhere differentiable and $f'(x)=0$ for every point of differentiability. The image of this set is $[0,1]$ from continuity and the fact that $f(\mathcal{C})=[0,1]$ and therefore the measure is not null. I note that this is also the statement in Frank Jones Lebesgue Integration on Euclidean Space. (unless I am mistaken). What am I missing?