 # Step Function: Simple Definition, Examples

Share on Example of a step function (the red graph). This particular step function is right-continuous.

A step function features constant or “flat” areas punctuated by abrupt jumps, called jump discontinuities. It looks exactly like what the name suggests: it’s a series of one or more steps. More technically, it’s defined as a piecewise function, with a finite number of pieces.

## Formal Definition

Formally, it is defined as a function over the real numbers that can be written as a finite linear combination of indicator functions of intervals.

It can also be defined as a function over the real numbers that can be written as: Where:

• n is an integer greater than zero,
• αi are real numbers.
• Ai are intervals,
• ΧA is the indicator function, defined as: ## Simple Examples of the Step Function

By the above definition, the constant function f(x) = A is a step function, though there is only one interval. In terms of “steps”, there’s just one; Imagine you’re standing on the first step of a set of stairs. There’s no up or down—just the flat surface you’re standing on. Another very simple step function—with a step in it—is the sign function. This function sends all positive numbers to +1, and all negative numbers to -1, so it has just one jump discontinuity. The sign function.

The rectangular function is constant at 0 for most values of x, but sends one interval Ai to a constant a. A rectangular function.

The Heaviside function is very like the sign function in that it has just discontinuity, at zero, but it sends all positive numbers to 1 and all negative numbers to zero. 