Step function

In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

Contents

Definition and first consequences

A function f: \mathbb{R} \rightarrow \mathbb{R} is called a step function if it can be written as

f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\, for all real numbers x

where n\ge 0, \alpha_i are real numbers, A_i are intervals, and \chi_A\, is the indicator function of A:

\chi_A(x) =
\begin{cases}
1 & \mbox{if } x \in A, \\
0 & \mbox{if } x \notin A. \\
\end{cases}

In this definition, the intervals A_i can be assumed to have the following two properties:

  1. The intervals are disjoint, A_i\cap A_j=\emptyset for i\ne j
  2. The union of the intervals is the entire real line, \cup_{i=0}^n A_i=\mathbb R.

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function

f = 4 \chi_{[-5, 1)} %2B 3 \chi_{(0, 6)}\,

can be written as

f = 0\chi_{(-\infty, -5)} %2B4 \chi_{[-5, 0]} %2B7 \chi_{(0, 1)} %2B 3 \chi_{[1, 6)}%2B0\chi_{[6, \infty)}.\,

Examples

Non-examples

Properties

See also

References

  1. ^ for example see: Bachman, Narici, Beckenstein. "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8. 
  2. ^ Weir, Alan J. "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.