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:

and, for a compact interval,

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.

Let ${\displaystyle I}$ be an interval in the real line ${\displaystyle \mathbb {R} }$. A function ${\displaystyle f\colon I\to \mathbb {R} }$ is absolutely continuous on ${\displaystyle I}$ if for every positive number ${\displaystyle \varepsilon }$, there is a positive number ${\displaystyle \delta }$ such that whenever a finite sequence of pairwise disjoint sub-intervals ${\displaystyle (x_{k},y_{k})}$ of ${\displaystyle I}$ with ${\displaystyle x_{k}<y_{k}}$ satisfies

then

The collection of all absolutely continuous functions on ${\displaystyle I}$ is denoted ${\displaystyle \operatorname {AC} (I)}$.

The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:

If these equivalent conditions are satisfied, then necessarily any function g as in condition 3. satisfies g = f ′ almost everywhere.