Absolute continuity

Lipschitz continuous ⊆ absolutely continuous ⊆ uniformly continuous ⊆ continuous

and, for a bounded interval,

absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere.

Absolute continuity of functions

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.

Definition

Let ${\displaystyle I}$ be an interval in the real line ${\displaystyle \mathbb{R}}$. A function ${\displaystyle f:I\to \mathbb{R}}$ is absolutely continuous on ${\displaystyle I}$ if for every positive number ${\displaystyle \epsilon }$, 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

${\displaystyle \sum _{k=1}^N(y_k-x_k)<\delta }$

then

${\displaystyle \sum _{k=1}^N|f(y_k)-f(x_k)|<\epsilon .}$

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

Equivalent definitions

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

1. f is absolutely continuous;
2. f has a derivative f′ almost everywhere, the derivative is Lebesgue integrable, and $${\displaystyle f(x)=f(a)+{\int }_a^x{f}^{\prime }(t)dt}$$ for all x on [a,b];
3. there exists a Lebesgue integrable function g on [a,b] such that $${\displaystyle f(x)=f(a)+{\int }_a^{x}g(t)dt}$$ for all x in [a,b].

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

Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.

For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity.

Properties

• The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a (bounded) closed interval, then their product is also absolutely continuous. On unbounded intervals this may fail: for example on ${\displaystyle \mathbb{R}}$, the function ${\displaystyle f(x)=x}$ is absolutely continuous, but ${\displaystyle f(x{)}^2}$ is not even uniformly continuous.
• If an absolutely continuous function f is defined on a (bounded) closed interval and is nowhere zero then 1/f is absolutely continuous.
• Every absolutely continuous function (over an interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous.
• If f: [a,b] → R is absolutely continuous, then it is weakly differentiable; conversely if f: [a,b] → R is weakly differentiable, then it coincides almost everywhere with an absolutely continuous function; this provides a characterization of Sobolev spaces on intervals of the real line.
• If f: [a,b] → R is absolutely continuous, then it is of bounded variation on [a,b].
• If f: [a,b] → R is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on [a,b].
• If f: [a,b] → R is absolutely continuous, then it has the Luzin N property (that is, for any ${\displaystyle N\subseteq [a,b]}$ such that ${\displaystyle \lambda (N)=0}$, it holds that ${\displaystyle \lambda (f(N))=0}$, where ${\displaystyle \lambda }$ stands for the Lebesgue measure on R).
• f: [a,b] → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property. This statement is also known as the Banach-Zareckiǐ theorem.
• If f: I → R is absolutely continuous and g: R → R is globally Lipschitz-continuous, then the composition g ${\displaystyle \circ }$ f is absolutely continuous. Conversely, for every function g that is not globally Lipschitz continuous there exists an absolutely continuous function f such that g ${\displaystyle \circ }$ f is not absolutely continuous.

Examples

The following functions are uniformly continuous but not absolutely continuous:

• The Cantor function on [0, 1] (it is of bounded variation but not absolutely continuous);
• The function:$${\displaystyle f(x)=\{\begin{array}{ll}0,& \text{if }x=0\\ x\sin (1/x),& \text{if }x\ne 0\end{array}}$$ on a finite interval containing the origin.

The following functions are absolutely continuous but not α-Hölder continuous:

• The function f(x) = x^β on [0, c], for any 0 < β < α < 1

The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous:

• The function f(x) = √x on [0, c], for α ≤ 1/2.

Generalizations

Let (X, d) be a metric space and let I be an interval in the real line R. A function f: I → X is absolutely continuous on I if for every positive number ${\displaystyle \epsilon }$, there is a positive number ${\displaystyle \delta }$ such that whenever a finite sequence of pairwise disjoint sub-intervals [x_k, y_k] of I satisfies:

${\displaystyle \sum _k\left|y_k-x_k\right|<\delta }$

then:

${\displaystyle \sum _{k}d\left(f(y_k),f(x_k)\right)<\epsilon .}$

The collection of all absolutely continuous functions from I into X is denoted AC(I; X).

A further generalization is the space AC^p(I; X) of curves f: I → X such that:

${\displaystyle d\left(f(s),f(t)\right)\le {\int }_s^{t}m(\tau )d\tau \text{ for all }[s,t]\subseteq I}$

for some m in the L^p space L^p(I).

Properties of these generalizations

• Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.
• If f: [a,b] → X is absolutely continuous, then it is of bounded variation on [a,b].
• For f ∈ AC^p(I; X), the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ L^p(I; R) such that:$${\displaystyle d\left(f(s),f(t)\right)\le {\int }_s^{t}m(\tau )d\tau \text{ for all }[s,t]\subseteq I.}$$