Elementary question about real analysis Suppose $A \subseteq \mathbb{R}$, and let $x \in [a,b]$. Put

$$ F(x) = \int_{[a,x]} \chi_A dm $$

Then of course we know that $F $ is differentiable for almost all $x \in [a.b]$ and 
$$ F'(x) = 1  \; \; \text{for almost all } x \in [a,b] \cap A $$

Can someone explain to me why does it follow that $F'(x) = 1 $ implies that there exists $\epsilon$ and $\delta &gt;0$ such that 

$$ 1 \geq \frac{ m( [x , x +h] \cap A )}{h} &gt; 1 - \epsilon \; \; for\; \; 0 &lt; h &lt; \delta$$

and 

$$ 1 \geq \frac{ m( [x , x +k] \cap A )}{k} &gt; 1 - \epsilon \; \; for \; \; 0 &lt; k &lt; \delta$$
$$F(x+h)-F(x)=\int_{[x,x+h]}\chi_{A}dm=m([x,x+h]\cap A)$$
This is not true. If $A$ is the Cantor set, then $F=0$ and so $F'=0$ everywhere.