Absolute continuity
In calculus, 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, expressed by the fundamental theorem of calculus in the framework of Riemann integration. Such generalizations are often 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:
and:
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 [0, π/2), 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.
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