# Absolute continuity

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.

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-Nikdoym 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.