Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous. 
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$ 
\Big | \frac{1}{|I\_{1}|}\int\_{I\_{1}}f'(x)dx- \frac{1}{|I\_{2}|}\int\_{I\_{2}}f'(x)dx\Big|\overset{|J|\rightarrow 0}{\longrightarrow} 0,
$$
for any $I\_{1}\cap I\_{2}=\emptyset$ and $I\_{1}, I\_{2}\subset J$ intervals.



Let f:J→R be an absolutely continuous and f'\in...?
Modification after the comments from below.


Let I\_1=(a,b) and I\_2=(c,d), where a< b<=c< d. Then
\frac{1}{|I\_1|}\int\_{I\_1}f'(t)dt=\frac{f(b)-f(a)}{b-a}:=J\_1, 
similarly,
\frac{1}{|I\_2|}\int\_{I\_2}f'(t)dt=\frac{f(d)-f(c)}{d-c}:=J\_2. 
Let |J| \to 0,
J\_1 \to f(b-), while J\_2 \to f(b+). 
So one natural assumption is that f' exists everywhere! 


