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:

    ... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.
    George Eliot [Mary Ann (or Marian)

    [As teenager], the trauma of near-misses and almost- consequences usually brings us to our senses. We finally come down someplace between our parents’ safety advice, which underestimates our ability, and our own unreasonable disregard for safety, which is our childlike wish for invulnerability. Our definition of acceptable risk becomes a product of our own experience.
    Roger Gould (20th century)