## 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 [*x*_{k}, *y*_{k}], *k* = 1, ..., *n* satisfies then 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* = *d*μ/*d*ν, 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. |