I was wondering why the definition of absolute continuity of a function involves multiple intervals:
A function $f: E \to \mathbb{R}$ is absolutely continuous on an interval $E$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x_k, y_k)$ of $E$ satisfies
$$ \sum_{k} |y_{k} - x_{k}| < \delta$$
then
$$\sum_{k} |f(y_{k}) - f(x_{k})| < \epsilon$$
Wouldn't it be equivalent if we replaced the finite sequence of pairwise disjont sub-intervals with just a single interval, like this?:
A function $f: E \to \mathbb{R}$ is absolutely continuous on an interval $E$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that whenever an interval $(x_k, y_k)$ of $E$ satisfies
$$ |y_{k} - x_{k}| < \delta$$
then
$$|f(y_{k}) - f(x_{k})| < \epsilon$$
Obviously when $f$ is absolutely continuous it also satisfies my proposed 'definition', but why isn't the converse true?
Your proposed definition is just uniform continuity. There exists a function, which is uniformly continuous but not absolutely continuous. For details see Cantor function.