Absolute continuity - Imported Knowlpedia Content

A function $f:[a,b]\to\mathbb{R}$ is absolutely continuous if for every $\varepsilon>0$ there exists $\delta>0$ such that for every finite collection of pairwise disjoint subintervals $(a_k,b_k)\subseteq [a,b]$ with $\sum_k (b_k-a_k)<\delta$ , one has

\sum_k |f(b_k)-f(a_k)|<\varepsilon.

Remarks

Absolute continuity is stronger than uniform continuity and implies bounded variation. A key characterization is that $f$ is absolutely continuous if and only if there exists a Lebesgue integrable function $g$ on $[a,b]$ such that $f(x)=f(a)+\int_a^x g(t)\,dt$ for all $x\in[a,b]$ (with the integral taken as the Lebesgue integral); in that case the derivative satisfies $f'(x)=g(x)$ almost everywhere.

Examples

If $g$ is Lebesgue integrable on $[a,b]$ and $f(x)=\int_a^x g(t)\,dt$ , then $f$ is absolutely continuous.
Every Lipschitz continuous function on $[a,b]$ is absolutely continuous.
The Cantor function is continuous and of bounded variation but not absolutely continuous; it increases on a set of measure zero.