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 Georg Aumann in 1954; the corresponding regulated integral was promoted by the Bourbaki group, including Jean Dieudonné.

Let X be a Banach space with norm || - ||_X. A function f : [0, T] → 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):

It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:

Let Reg([0, T]; X) denote the set of all regulated functions f : [0, T] → X.

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