absolute continuity
Absolute continuity
Absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. It's essentially just a continuous function with extra constraints so that it is guaranteed to have a derivative almost everywhere, and its derivative can be integrated to return the original function. It allows one to obtain generalizations of the relationship differentiation and integration using the fundamental theorem of calculus. The continuous functions that we normally encounter are most likely also absolute continuous.
The intuition for absolute continuity is similar to uniform continuity. The difference is that uniform continuity (over an interval) only requires that there exist a suitable \[\delta\] corresponding to an \[\epsilon\] regardless of where in \[\left[ a,b \right]\] the points \[x\] and \[y\] are located, while absolute continuity goes further by considering disjoint intervals collectively rather than just pairs of points. It demands that there exist a suitable \[\delta\] corresponding to an \[\epsilon\] when we split up the \[\delta\]-sized interval into finitely many disjoint "tiny intervals" whose lengths sum to \[<\delta\] and the corresponding "tiny epsilons" sum to \[<\epsilon\].
A simple way to think of absolute continuity is that it tells us that the image under \[f\] of a sufficiently small finite collections of intervals is arbitrarily small ("small" here refers to total length).
Definition
\[f\] is absolutely continuous on \[\left[ a,b \right]\] if and only if for each \[\epsilon>0\] there exists a \[\delta>0\] such that for each \[n\in \mathbb{N}\], \[a\le x_{1}<y_{1}\le x_{2}<y_{2}\le\dots\le x_{n}<y_{n}\le b\] with \[\sum_{i=1}^{n}(y_{i}-x_{i})<\delta\] whenever \[\sum_{i=1}^{n}\left| f(y_{i})-f(x_{i}) \right|<\epsilon\].
An equal definition would be, \[f\] is absolutely continuous if and only if \[f\] is differentiable almost everywhere and \[f(x)=f(a)+\int_{a}^{x}f^{\prime}(t)\,dt\] for all \[x\in \left[ a,b \right]\].