In mathematics, a function on the real numbers is called a step 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 ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$ is called a step function if it can be written as ^{[citation needed]}

where ${\displaystyle n\geq 0}$, ${\displaystyle \alpha _{i}}$ are real numbers, ${\displaystyle A_{i}}$ are intervals, and ${\displaystyle \chi _{A_{i}}}$ is the indicator function of ${\displaystyle A_{i}}$:

In this definition, the intervals ${\displaystyle A_{i}}$ can be assumed to have the following two properties:

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

Sometimes, the intervals are required to be right-open or allowed to be singleton. The condition that the collection of intervals must be finite is often dropped, especially in school mathematics, though it must still be locally finite, resulting in the definition of piecewise constant functions.