Absolutely continuous functions and general absolute continuity First, the definitions:
$f$ is AC on $E$ if $$\forall \epsilon >0\ \exists \delta >0\ \forall \{[a_k,b_k]\}_{k=1}^N \mbox{ such that }a_k,b_k \in E,\ \Sigma(b_k - a_k) <\delta : \Sigma| f(b_k) - f(a_k)| <\epsilon.$$ 
$f$ is GAC on $E$ if $E= \underset{n \in \mathbb{N}} {\biguplus} E_n$, $f$ is AC on $E_n\ \forall n$ and continuous on $E$.

My question is to prove that if $f$ is GAC on every measurable $E \subset I=[a,b]$ then $f$ AC on $I$. 

I tried using the fact that it's GAC on $I$, and concluding that it's bounded variation on every $E_n$, thus can be written as two monotone functions, and eventually write $f$ as the sum of two (almost-lacking continuity) GAC monotone functions. Every monotone and GAC function is also AC. How can I use what's given to me and get continuity?? Any suggestions will be much appreciated.
 A: I'm quite sure that John is wrong... as Umberto P. mentioned earlier, GAC on [a,b] implies GAC on every borel subset (take $[a,b]=\biguplus E_n$, $E\in B([a,b])$ s.t. $f$ is AC on each $E_n$, then it will also be AC on each $E\cap E_n$). So, the claim actually says that GAC on a segment (or whatever) implies AC. Which is absurd (take $x\sin(\frac{1}{x})$).
A: StackExchange, as currently set up, always provides a list of suggestions for related questions.  How tempting to sample these ... except sometimes you are thrown into a 2013 mess as here.  The accepted answer (not upvoted however) points out that the question is absurd with an easy counterexample. 
The question itself is also badly confused.  There is no mention that the intervals $\{[a_k,b_k]\}$ are to be nonoverlapping, although most of us would take that for granted anyway.  There is the strange insistence that the condition to be posed is that "$f$ is GAC on every measurable set $E\subset [a,b]$."  As the answer points out this condition is hereditary in any case.  And what does "measurable" have to do with it?  Then the final absurdity is that, while there are a host of interesting problems connected with these definitions, the one problem posed here is just flat out wrong!
If you end up looking at this stuff, please immediately turn to Chapter VII of Saks, Theory of the Integral.  He defines VB (variation bornée),
VBG (variation bornée généralisée), AC (absolument continues),
and ACG (absolument continues généralisées). The original text was in French and the translator (L.C. Young) chose to preserve the word order of that language for the VB and ACG concepts.
Out of respect for Saks and the many researchers who have employed these concepts please do not use GAC to describe what we all call ACG! [Save GAC for "get a clue" or "guilty as charged" etc.]
There are many ideas in Saks that can be used to pose problems for graduate students.  Here are a few nice ones, unlike the rubbish in this question.

  
*
  
*Show that an everywhere differentiable function $F$ on an interval $[a,b]$ must be ACG there, but not necessarily of bounded variation.
  
*Show that a continuous function on a closed set $E$ is VBG (or ACG) on $E$ if and only if there is a portion of $E$ on which $F$ is
  VB (or AC).
  
*Show that a function which is ACG on a set necessarily satisfies Lusin's condition $N$ on that set.
  
*Show that a function $F$ which is ACG on an interval $[a,b]$ and has almost everywhere a nonnegative upper right Dini derivative
  ${D}^+F(x)\geq 0$, is nondecreasing.
  
*Show that a continuous function $F$ on a closed set $E$ is ACG if and only if it is ACG on every subset of $E$ of measure zero.

