> [!definition] > > Let $F: \real \to \complex$. $F$ is **absolutely continuous**, if for every $\eps > 0$, there exists $\delta > 0$ such that for any finite set of disjoint intervals $\seqf{(a_j, b_j)}$, > $ > \sum_{j = 1}^n(b_j - a_j) \Rightarrow \sum_{j = 1}^n \abs{F(b_j) - F(a_j)} < \eps > $ > [!theorem] > > Let $F: [a, b] \to \complex$, then the following are equivalent: > 1. $F$ is absolutely continuous on $[a, b]$. > 2. $F(x) - F(a) = \int_a^x f(t)dt$ for some $f \in L^1([a, b])$. > 3. $F$ is differentiable a.e. on $[a, b]$ with $F' \in L^1([a, b], m)$, and $F(x) - F(a) = \int_a^x F'(t)dt$. > [!theorem] > > Let $F \in AC([a, b])$, then $\abs{F} \in AC([a, b])$. If $F \in C^1([a, b])$, then $F \in AC([a, b])$, and $\abs{F} \in AC([a, b])$ where $F'$ is a.e. equal to its weak derivative.