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 ${\displaystyle I}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle f\colon I\to \mathbb {R} }$ ${\displaystyle f\colon I\to \mathbb {C} }$

• has both a left and a right limit at each point and ${\displaystyle x\in {]a,b[}}$
• in the case in a right-side limit value and in the case in has a left-hand limit. ${\displaystyle a\in I}$ ${\displaystyle a}$ ${\displaystyle b\in I}$ ${\displaystyle b}$

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. ${\displaystyle a_{n}}$ ${\displaystyle \lim f(a_{n})\neq f(\lim a_{n})}$

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

Examples

The sign function is an example of a control function with a jump point.

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 . ${\displaystyle 0}$
• 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 . ${\displaystyle \sin({\tfrac {1}{x}})}$ ${\displaystyle 0}$
• 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 ${\displaystyle I}$ ${\displaystyle {\mathcal {R}}(I)}$

${\displaystyle \|f\|_{\infty }=\sup _{x\in I}|f(x)|}$

is a Banach space. With the (point-wise) product of two rule functions, it is even a Banach algebra . ${\displaystyle {\mathcal {R}}(I)}$

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 ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle f\colon [a,b]\to \mathbb {C} }$ ${\displaystyle (h_{n})}$

${\displaystyle \lim _{n\to \infty }\|f-h_{n}\|_{\infty }=0}$

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. ${\displaystyle \|{\cdot }\|_{\infty }}$

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 ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle (h_{n})}$ ${\displaystyle \|f-h_{n}\|_{\infty }\to 0}$ ${\displaystyle \|{\cdot }\|_{\infty }}$

${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x:=\lim _{n\to \infty }\int _{a}^{b}h_{n}(x)\mathrm {d} x}$

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

1. ↑ control function . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. ^ Konrad Königsberger : Analysis 1 . Springer-Verlag, Berlin et al., 2004, ISBN 3-540-41282-4 , p. 193.
3. ^ Martin Barner, Friedrich Flohr: Analysis I. 4th edition. de Gruyter, Berlin 1991, pages 342-343.
4. ^ Martin Barner, Friedrich Flohr: Analysis I. 4th edition. de Gruyter, Berlin 1991, page 340.