Different definition of continuity

Question

Condition: $f:I\to\mathbb R$ is continuous. For any countable sequence of pairwise disjoint sub-intervals $(x_k, y_k)$ of $I$, we have $\forall\epsilon\exists\delta$ such that $$ \sum_{k} |y_{k} - x_{k}| < \delta$$ implies $$\sum_{k} |f(y_{k}) - f(x_{k})| < \epsilon.$$

Is this condition a necessary or sufficient condition of absolution continuity? Note that the order of the logic identifiers has changed.

A function $f: I \to \mathbb{R}$ is absolutely continuous on an interval $I$ 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 $I$ satisfies $$ \sum_{k} |y_{k} - x_{k}| < \delta$$ then $$\sum_{k} |f(y_{k}) - f(x_{k})| < \epsilon$$

Answer 1

Given any function $f:I\to\mathbb R$ (not necessarily continuous), the condition: For any countable sequence of pairwise disjoint sub-intervals $(x_k, y_k)$ of $I$, we have $\forall\epsilon\exists\delta$ such that $$ \sum_{k} |y_{k} - x_{k}| < \delta$$ implies $$\sum_{k} |f(y_{k}) - f(x_{k})| < \epsilon$$ is trivially true.

Proof: Given any countable sequence of pairwise disjoint sub-intervals $(x_k, y_k)$ of $I$, just choose $\delta = \frac{1}{2} \sum_{k} |y_{k} - x_{k}|$. Then the condition $ \sum_{k} |y_{k} - x_{k}| < \delta$ will be false and so the implication "$ \sum_{k} |y_{k} - x_{k}| < \delta$ implies $\sum_{k} |f(y_{k}) - f(x_{k})| < \epsilon$" will be trivially true.

So this conditionis not sufficient for absolute continuity or even continuity.

Answer 2

Yes, they are equivalent. Suppose you choose $\delta$ according to the usual definition of absolute continuity with $\epsilon$ repalced by $\epsilon /2$. If $(a_k.b_k)$ is a disjoint sequence of interval with total length less than $\delta$ then $\sum\limits_{k=1}^{N} |f(b_k)-f(a_k)| < \epsilon /2$ for each $N$. Let $N \to \infty$ to complete the proof.