Control function
In mathematics, a control function or step-continuous function is a function whose only points of discontinuity are step points. They play an important role in integration theory . The term "regulation function" ( fonction réglée ) was introduced by the French mathematicians school.
definition
Let it be an open, half-open or closed interval with a starting point and an end point . A real or complex valued function or is called a control function, if it
- has both a left and a right limit at each point and
- in the case in a right-side limit value and in the case in has a left-hand limit.
Since the left-hand and right-hand limit values do not have to match, a control function can have jump points, that is, points where there is a sequence for which applies. Control functions are therefore also referred to as continuous functions . A control function is called piecewise continuous if it only has a finite number of places at which it is not continuous , and thus only has a finite number of jumps.
The definition can be generalized by considering Banach space valued functions instead of real or complex valued functions.
Examples
- Control functions
- Every continuous function on an interval is a control function without jumps.
- The Heaviside function and the sign function are control functions with a jump at the point on an interval around the zero point .
- Every real-valued monotonic function on an interval is a control function.
- The Thoma's function is a control function with a countable number of jump points. It is therefore not piecewise continuous.
- No control functions
- A function with a pole within the observed interval is not a control function, because at this point at least one of the limit values only exists as an improper limit value .
- The function is not a control function in any interval that contains the zero point, because it has no limit value at this point .
- The Dirichlet function is not a control function, because there is no limit value at any point. It has an uncountable number of jump points.
properties
characterization
A function is jump-continuous if and only if it does not have any points of discontinuity of the second kind . Every 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 shows.
Spaces of control functions
The set of control functions on an interval form a vector space , which is denoted by. With the supremacy norm
is a Banach space. With the (point-wise) product of two rule functions, it is even a Banach algebra .
Approximability
Every control function on a compact interval can be approximated uniformly by a sequence of step functions . This means that for every control function or a sequence of step functions exists so that
applies, where is the supremum norm. Conversely, every function on a compact interval that can be uniformly approximated by step functions is a rule function. Therefore, this property can be used as an alternative to step continuity to define control functions.
Integral of control functions
Let be a rule function and a sequence of step functions with , where the supremum norm is. Then an integral can through
To be defined. This integral is generalized by the Riemann integral .
literature
- Herbert Amann, Joachim Escher: Analysis II. Birkhäuser, Basel 1999, p. 4.
Individual evidence
- ↑ control function . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
- ^ Konrad Königsberger : Analysis 1 . Springer-Verlag, Berlin et al., 2004, ISBN 3-540-41282-4 , p. 193.
- ^ Martin Barner, Friedrich Flohr: Analysis I. 4th edition. de Gruyter, Berlin 1991, pages 342-343.
- ^ Martin Barner, Friedrich Flohr: Analysis I. 4th edition. de Gruyter, Berlin 1991, page 340.