Staircase Function

Definition

A function that is a piecewise constant function is called a step function.

Description

As shown in the figure above, it looks like a staircase, hence the name step function. It is also known as the Heaviside function, named after Heaviside, who is known to be the first to propose it. Heaviside was the person who created a method for solving differential equations in electrical circuits, which is the Laplace transform. A typical example of a step function is the Gaussian function.

It is commonly denoted by $H, h$, following Heaviside’s name. Unless specifically mentioned, referring to a step function usually means the unit step function described below.

Unit Step Function

The following $H$ is called the unit step function.

$$ H(x) = \begin{cases} 1 & x \gt 0 \\ 0 & x \le 0\end{cases} $$