Regulated function

In mathematics, a regulated function (or ruled function) is a "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.

Definition

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:

Properties of regulated functions

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

http://math.stackexchange.com/questions/84870/how-to-show-that-a-set-of-discontinuous-points-of-an-increasing-function-is-at-m

References


External links

http://math.stackexchange.com/questions/84870/how-to-show-that-a-set-of-discontinuous-points-of-an-increasing-function-is-at-m

http://math.stackexchange.com/questions/584735/bounded-variation-functions-have-jump-type-discontinuities?rq=1

https://math.stackexchange.com/questions/112067/how-discontinuous-can-a-derivative-be

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