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.

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

where ${\displaystyle n\geq 0,}$ ${\displaystyle \alpha _{i}}$ are real numbers, ${\displaystyle A_{i}}$ are intervals, and ${\displaystyle \chi _{A}\,}$ (sometimes written as ${\displaystyle 1_{A}}$) is the indicator function of ${\displaystyle A}$:

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