Absolute continuity

Remarks

Example: a Lipschitz function with constant $L > 0$ is absolutely continuous. Given $\varepsilon$, take $\delta = \varepsilon / L$; then $\sum_k \abs{f(b_k) - f(a_k)} \le L \sum_k (b_k - a_k) < \varepsilon$.

Non-example: the Cantor function $c$ on $[0, 1]$ is continuous but not absolutely continuous. For each $\delta > 0$, the Cantor set is covered by finitely many disjoint open intervals of total length less than $\delta$. The changes of $c$ across these intervals add up to $c(1) - c(0) = 1$.

Equivalent form: $f$ is absolutely continuous on $[a, b]$ if and only if there is a Lebesgue integrable $g$ on $[a, b]$ with $f(x) = f(a) + \int_a^x g(t)\, dt$ for all $x$. Then $f' = g$ almost everywhere. With $n = 1$ only, the definition reduces to uniform continuity.