Step function (real function)

This article covers the stair functions on . For step functions that are used, for example, in the construction of the Lebesgue integral, see Simple function . ${\displaystyle \mathbb {R} }$

Example of a step function

In mathematics, a step function is a special real function that only takes on a finite number of function values ​​and is constant in parts. This gives the function graph of a staircase function its characteristic and name-giving appearance, which is similar to an ascending and descending staircase.

definition

One function

${\displaystyle f\colon [a;b]\to \mathbb {R} }$

is called a step function if there are numbers with ${\displaystyle t_{0},t_{1},\dotsc ,t_{n}}$

${\displaystyle a=t_{0}<t_{1}<t_{2}<\dotsb <t_{n}=b}$

there and numbers so that ${\displaystyle c_{1},c_{2},\dotsc ,c_{n}}$

${\displaystyle f(x)=c_{i}{\text{ für alle }}x\in (t_{i-1},t_{i})}$

and all applies. The function values at the "support points" are arbitrary, but real. ${\displaystyle i=1,\dotsc ,n}$ ${\displaystyle f(t_{i})}$

use

Step functions is also used for approximation of integrals . The integral of a step function is given by

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=\sum _{i=1}^{n}c_{i}\ell (I_{i})}$

Are defined. The advantage here is that you can do without a limit value process and only have finite sums . In the empirical formula denotes the value of on the interval and the length of this interval, ie . ${\displaystyle c_{i}}$ ${\displaystyle f}$ ${\displaystyle I_{i}=(t_{i-1},t_{i})}$ ${\displaystyle \ell (I_{i})}$ ${\displaystyle t_{i}-t_{t-i}}$

The simple definition of the integral of a step function already gives you a powerful mathematical aid: every constrained, continuous function with can be approximated as precisely as you want using a step function . So the integral of this function can also be approximated with any precision. This fact is an important foundation for the definition of the Riemann integral . In this way, Jean Gaston Darboux simplified the introduction of the Riemann integral. ${\displaystyle f\colon D\rightarrow Y}$ ${\displaystyle D,Y\subseteq \mathbb {R} }$

Examples

• The Heaviside function is 0 for every negative number, otherwise 1.
• Rectangular function
• Rounding function and rounding function

Demarcation

The stair functions are very similar to both the simple functions and the jump functions , but should not be confused with them.

For example, simple functions only take on a finite number of values, but can still be much more complex, since they are not defined via intervals on the basic space, but via measurable quantities . For example, the Dirichlet function is a simple function, but not a step function in the sense mentioned here, as it has an uncountable number of jump points and is not constant in any interval, however small. In addition, simple functions are defined on any measuring room , whereas staircase functions are only defined on. However, every step function is always a simple function. ${\displaystyle \mathbb {R} }$

Like the step functions, the jump functions are also defined on the real numbers. However, they always grow monotonically , but they can also have many jump points that can be counted.

generalization

A stochastic generalization of a staircase function is an elementary, predictable stochastic process . It plays a similar role in the construction of the Ito integral as the simple functions in the construction of the Lebesgue integral .

Web links

• Eric W. Weisstein : Step Function . In: MathWorld (English).

Individual evidence

1. ^ Otto Forster : Analysis 1 . Differential and integral calculus of a variable. 11th, expanded edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00316-6 , p. 105 , doi : 10.1007 / 978-3-658-00317-3 .