Step Functions

Example 1:

Let the function shown be defined for all the integers as

$\begin{array}{l}y=-2\text{for}x<1\\ y=3\text{for}x\ge 1\end{array}$

This function is made up of infinitely many discrete points each of which have a $y$ -coordinate of either $-2$ or $3$ .

Example 2:

Consider the function, $y=-n\text{for}n<x\le n+1$ where $x\in ℝ,n\in \mathbb{Z}$ .

This function is made up of infinitely many horizontal segments, each of which includes the right endpoint but not the left endpoint.

Example 3:

$\begin{array}{l}y=-1\text{for}x<0\\ y=0\text{for}x=0\\ y=1\text{for}x>1\end{array}$ .

This step function is called the Heaviside Function. It may look very simple, but it is extremely important in electrical engineering and control theory.