I believe this answers the question:
MR0425042 (54 #13000)
Goffman, Casper
A bounded derivative which is not Riemann integrable.
Amer. Math. Monthly 84 (1977), no. 3, 205--206.
In 1881 Volterra constructed a bounded derivative on $[0,1]$ which is not Riemann integrable. Since that time, a number of authors have constructed other such examples. These examples are generally relatively complicated and/or involve nonelementary techniques. The present author provides a simple example of such a derivative $f$ and uses only elementary techniques to show that $f$ has the desired properties.
The paper is available here:
Yes, we can do this by a small modification of the original argument, see the linked question. I'll describe the whole argument again here, but if you already read those answers, then the short version is that we remove successively less of each $I_n(p)$ as we approach the boundary of that interval. This allows us (for each $x\in I_n(p)$) to replace $I_n(p)$ by its maximal subinterval centered at $x$ while only slightly disturbing the approximate densities.
Construct open sets $G_1\supset G_2 \supset \ldots \supset E$, as follows. Start with any open $G_1\supset E$, and write $G_1 = \bigcup I_1(p)$ as the union of its components. For each $I_1(p)$, we construct an open set $G_2(p)$ with $E\subset G_2(p)\subset I_1(p)$. This $G_2(p)$ will cover only a small portion of $I_1(p)$ (possible since $|E|=0$), and, moreover, this portion will not be unduly concentrated near the boundary of $I_1(p)$ (this is the modification). More specifically, if $I_1(p)=(a,b)$, then we demand that
$$
|(a,a+h)\cap G_2(p)| \le 2^{-1}h \quad\quad\quad (1)
$$
for all $h>0$, and similarly near $b$. We can do this by cutting $I_1(p)$ into pieces $(a+2^{-k}, a+2^{-k+1})$ and treating these separately (if an endpoint is in $E$ here, I very slightly move such a point).
Then we put $G_2=\bigcup G_2(p)=\bigcup I_2(q)$. We repeat this whole procedure, with $2^{-1}$ replaced by $2^{-2}$ in (1), to obtain $G_3$ etc., and we finally set $F=\bigcup (G_{2n-1}\setminus G_{2n})$ (start with $G_1$, remove $G_2$, add $G_3$ etc. etc.).
If $x\in E$, then $x\in I_n(p_n)$ for some (unique) $p_n$ for all $n$. If $n$ is odd and we write $I_n(p_n)=(a,b)$ and $x$ is closer to $a$ than to $b$, then we consider $J=(a,a+2(x-a))$. Then $J\cap F\supset J\setminus G_{n+1}$, so (1) says that $|J\cap F|/|J|$ is almost $1$ if $n$ is large.
Similarly, we find intervals for which this ratio is almost zero from the even $n$'s.
Best Answer
Having finite Dini derivatives implies that the function is continuous, and we even get differentiable a.e. (eg."Mean Value Theorems And Functional Equations")
Once we have continuity, we have integral formulas (eg."Recovering a Function from a Dini Derivative") in terms of upper/lower Riemman (but with a restricted partition)
and so since we have L1, we also get absolute continuity.