Absolute Continuity

Definition 1

• $(X, d)$
  • metric space
• $I := [a, b]$,
• $f: I \rightarrow X$,

\[\mathcal{J} := \{ ([a_{i}, b_{i}])_{i=1,\ldots,k} \mid a_{i} \le b_{i} \le a_{i + 1} \ k \in \mathbb{N} \} .\]

$f$ is said to be absolute continuous if for all $\epsilon > 0$, there exists $\delta > 0$ such that

\[([a_{i}, b_{i}])_{i = 1, \ldots, k} \in J, \ \sum_{i=1}^{k} \abs{ b_{i} - a_{i} } < \delta \Rightarrow \sum_{i=1}^{k} d(f(b_{i}) - f(a_{i})) < \epsilon .\]

We denote the collection of all absolute functions by $\mathrm{AC}(I; X)$.

■

Reference

• Absolute continuity - Wikipedia