′ 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. If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous. Every absolutely continuous function (over a compact 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 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 such that , it holds that , where stands for the Lebesgue measure on R). f: I → 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 ∘ 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 ∘ 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 on a finite interval containing the origin. The following functions are absolutely continuous but not α-Hölder continuous: the function f(x) = xβ on [0, c], for any The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous: the function f(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 , there is a positive number such that whenever a finite