Step Function - Definition and First Consequences

Definition and First Consequences

A function is called a step function if it can be written as

for all real numbers

where are real numbers, are intervals, and is the indicator function of :

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

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

  1. The intervals are disjoint, for
  2. The union of the intervals is the entire real line,

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


can be written as

Read more about this topic:  Step Function

Famous quotes containing the words definition and/or consequences:

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)

    There are more consequences to a shipwreck than the underwriters notice.
    Henry David Thoreau (1817–1862)