Regulated function

Under a control function or jump continuous function is understood in mathematics a function whose only discontinuities are jump points. They play an important role in the theory of integration. The term " control function " ( fonction réglée ) was introduced by the French mathematician school.

Definition

It is an open, semi-open or closed interval with start point and end point. A real - or complex-valued function and is called control function, if it

  • At each point of both a left -side and right-side has a limit and
  • In the case has a left- hand limit in a right-side limit and in the case.

Since the left - and right-sided limits do not necessarily match, a control function may have discontinuities, that is, points at which there is a sequence, applies. Control functions are therefore also known as jump continuous functions. A rule function is called while piecewise continuous if it has only finitely many points at which it is not continuous, and therefore has only finitely many jumps.

The definition can be generalized by instead of real or complex-valued functions considered Banach space - valued functions.

Examples

  • Every continuous function on an interval is a rule function without discontinuities.
  • The Heaviside function and the sign function are on an interval around zero control functions with a discontinuity at the point.
  • Each real-valued monotonic function on an interval is a control function.
  • The Thoma ash function is a control function with a countable number of discontinuities. Therefore, it is not piecewise continuous.
  • A function with a pole within the considered interval is no control function, because at this point there is at least one of the limits as improper limit.
  • The function is in no interval which includes the zero point, a control function, because they have no limit on the site.
  • The Dirichlet function is no control function, because you exist at any point in a limit. It has uncountably many discontinuities.

Properties

Characterization

A function is jump continuous if it has no discontinuities of the second kind. Each control function on a compact interval is limited. However, the reverse direction does not have to be true, as the example of the Dirichlet - function displays.

Spaces of control functions

The amount of control functions on an interval form a vector space which is denoted by. With the supremum norm

Is a Banach space. With the ( pointwise ) product of two control functions, these are even a Banach algebra.

Approximability

Each control function on a compact interval can be uniformly approximated by a sequence of step functions. That is, each control function and there exists a sequence of step functions, so that

Holds, where the supremum is. Conversely, each function on a compact interval, which can be uniformly approximated by step functions, a control function. Therefore, this property can be used alternatively to jump continuity to define control functions.

Integral control functions

Be a control function and a sequence of step functions with, where the supremum is. Then, by an integral

Be defined. This integral is generalized by the Riemann integral.

Thomae's function Nowhere continuous function Uniform norm
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