I don't seem to find version of this problem in the site, but I am sure this is pretty standard type of question.
$f$ be of bounded variation on $[0,1]$, and $f$ is absolutely continuous (AC) on $[\varepsilon,1]$ for all $\varepsilon >0$ and $f$ is continuous at $0$. Now the goal is to prove $f$ is absolutely continuous on whole interval $[0,1]$.
Moreover I am looking for a counterexample in the case when the bounded variation of $f $ is dropped.
Here is what I think,
Using continuity, I can find $\delta>0$ for given $\epsilon >0$ which bounds the sum in the definition of AC upto $\delta$. Then in the interval $[\delta,1]$, I can use given hypothesis. But I am not totally comfortable writing this rigorously.
For the counter example I can use $f(0)=0$ and $f(x)= x\sin (1/x)$ for $x$ not equal to $0$. I would love to see the rigorous proof and rigorous proof of counterexample. Thank you in advance.