I learned [this question][1] from math.stackchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does 
$$g(x):=\lim_{\epsilon\to 0}\frac{f(x+\epsilon)-f(x-\epsilon)}{2\epsilon}$$
exist for every $x\in[0,1]$ imply that $f$ is absolutely continuous? If the answer is "no", what if $g$ is additionally assumed being integrable or continuous? 

Moreover, if we do not know whether $f$ is of bounded variation or not, what can we say about the differentiability of $f$? For example, if $g\equiv 0$, will $f$ be a constant?

Any help is appreciated. 


  [1]: http://math.stackexchange.com/questions/235689/pointwise-existence-of-radon-nikodym-derivative-sufficient-for-absolute-continui