# A question about absolutely continuous functions

I know that the definition of absolute continuity is

for a real-valued function $$f$$, which is defined on $$[a,b]$$, if $$\forall\epsilon>0$$, $$\exists\delta>0$$, such that whenever $$\{(a_k, b_k)\}_{k=1}^{n}$$ are disjoint open intervals on $$[a,b]$$ and $$\sum\limits_{k=1}^{n}|a_k-b_k|<\delta$$, $$\sum\limits_{k=1}^{n}|f(a_k)-f(b_k)|<\epsilon$$.

So here is the question, if I replace the last condition with $$|\sum\limits_{k=1}^{n}(f(a_k)-f(b_k))|<\epsilon$$, does the function $$f$$ remain absolutely continuous?

I have tried to prove it or give a counterexample but all these efforts have failed. And I would like to appreciate it if you could help me solve this problem.

Just apply this new hypothesis to the subcollection of the intervals where $$f(b_k) \geq f(a_k)$$ and then to the subcollection of the intervals where $$f(b_k) < f(a_k)$$. You will see that $$f$$ is necessarily absolutely continuous.