Regulated function

In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Nicolas Bourbaki in 1949, in their book 'Livre IV: Fonctions d'une variable réelle'. In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Nicolas Bourbaki in 1949, in their book 'Livre IV: Fonctions d'une variable réelle'. Let X be a Banach space with norm || - ||X. A function f : → X is said to be a regulated function if one (and hence both) of the following two equivalent conditions holds true (Dieudonné 1969, §7.6):