If f is differentiable in $[0,1]$, let $E_0=\{x\in [0,1]: f'(x)=0\}$, how to prove $m(f(E_0))=0$($m$ is the Lebesgue measure)?
I tried to use some little open intervals to cover every points in $E_0$, but the number of points in $E_0$ is difficult to estimate, so I don't know how to continue.
Choose $x_0\in E_0$, then $|f(x)-f(x_0)|=o(|x-x_0|)$ follows. So, for every $k>0$, we can find a $\delta>0$ such that $|f(x)-f(x_0)|<\dfrac{1}{k}|x-x_0|$ whenever $|x-x_0|<\delta$. Then I cannot do.