GeezerVison

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 $f:ℝ\to ℝ$ is called a step function if it can be written as

$f\left(x\right)=\sum _{i=0}^n{\alpha }_i{\chi }_{A_i}\left(x\right)$, for all real numbers $x$

where $n\ge 0$, ${\alpha }_i$ are real numbers, $A_i$ are intervals, and ${\chi }_{A_i}$ is the indicator function of $A_i$:

${\chi }_{A_i}\left(x\right)=\{\begin{array}{cc}1& \text{if }x\in A_i\\ 0& \text{if }x\notin A_i\end{array}$

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

1. The intervals are pairwise disjoint: $A_i\cap A_j=\mathrm{\emptyset }$ for $i\ne j$
2. The union of the intervals is the entire real line: $\bigcup _{i=0}^n{A}_i=ℝ.$

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

$f=4{\chi }_{[-5,1)}+3{\chi }_{(0,6)}$

can be written as

$f=0{\chi }_{(-\infty ,-5)}+4{\chi }_{[-5,0]}+7{\chi }_{(0,1)}+3{\chi }_{[1,6)}+0{\chi }_{[6,\infty )}.$

• A constant function is a trivial example of a step function. Then there is only one interval, $A_0=ℝ.$
• The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
• The Heaviside function H(x), which is 0 for negative numbers and 1 for positive numbers, is equivalent to the sign function, up to a shift and scale of range ($H=(\mathrm{sgn}+1)/2$). It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.

• The rectangular function, the normalized boxcar function, is used to model a unit pulse.

• The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an algebra over the real numbers.
• A step function takes only a finite number of values. If the intervals $A_i,$ for $i=0,1,\dots ,n$ in the above definition of the step function are disjoint and their union is the real line, then $f\left(x\right)={\alpha }_i$ for all $x\in A_i.$
• The definite integral of a step function is a piecewise linear function.
• The Lebesgue integral of a step function $f={\sum }_{i=0}^n{\alpha }_i{\chi }_{A_i}$ is $\int fdx={\sum }_{i=0}^n{\alpha }_i\ell \left(A_i\right),$ where $\ell \left(A\right)$ is the length of the interval $A$, and it is assumed here that all intervals $A_i$ have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.^[7]
• A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.^[8] In this case, it is locally a step function (globally, it may have an infinite number of steps). Usually however, any random variable with only countably many possible values is called a discrete random variable, in this case their cumulative distribution function is not necessarily locally a step function, as infinitely many intervals can accumulate in a finite region.

• Crenel function
• Piecewise
• Sigmoid function
• Simple function
• Step detection
• Heaviside step function
• Piecewise-constant valuation