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Absolute continuity

In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: absolutely continuous ⊆ uniformly continuous continuous and, for a compact interval, continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere. A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over , x2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function, which is not differentiable anywhere). Or it may be differentiable almost everywhere and its derivative f ′ may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval). This happens for example with the Cantor function. Let be an interval in the real line . A function is absolutely continuous on if for every positive number , there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals of with satisfies then The collection of all absolutely continuous functions on is denoted .

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