This might probably be classed as a soft question. But I would be very interested to know the motivation behind the definition of an absolutely continuous function. To state "A real valued function $ f $ on $ [a,b] $ is absolutely continuous on said interval if $\forall $$\epsilon >0 \ $, $\exists\delta>0\ $ such that $$ \sum^n_{k=1}|f(b_k) -f(a_k)|< \epsilon $$
for every $n$ disjoint subintervals $ \ (a_k,b_k) $ of $ \ [a,b] $, $k=1,\ldots,n$, such that $ \sum^n_{k=1}|b_k -a_k|< \delta $. Why the use of the disjoint sub-intervals? What purpose do they serve? Somehow the definition didn't seem natural to me.
What I mean is just as the notion of uniform continuity is motivated by the definition of continuity itself or as the concept of compactness serves to generalise the notion of finiteness, how can one look at Absolute Continuity in this respect?