Absolutely continuous function

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-continuousfunction 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.