Absolute continuity of real functions
In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n satisfies
Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.
The Cantor function is continuous everywhere but not absolutely continuous.
Absolute continuity of measures
If μ and ν are measures on the same measure space (or, more precisely, on the same sigma-algebra) then μ is absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. One writes "μ << ν".
The Radon-Nikodym theorem states that if μ is absolutely continuous with respect to ν, and ν is σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, i.e., a measurable function f taking values in [0,∞], denoted by f = dμ/dν, such that for any measurable set A we have
The connection between absolute continuity of real functions and absolute continuity of measures
A measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function
is an absolutely continuous real function.