Step Functions and Greatest Integer Function

A step function is a piecewise function that is constant on each interval of its domain, so its graph looks like a staircase of horizontal segments. Each step has an open circle on one end and a closed circle on the other to show which endpoints are included.

The greatest integer function, written ⌊x⌋ or [x], gives the largest integer less than or equal to x. For example, ⌊4.9⌋ = 4, ⌊3⌋ = 3, and ⌊−2.1⌋ = −3. It is also called the floor function.

Plot horizontal segments at each integer value y = n for n ≤ x < n+1. Each segment has a closed circle on the left and an open circle on the right. The graph looks like a series of steps moving up and to the right.

Postage costs, parking garage fees, and income tax brackets are all step functions — the output stays constant within a range, then jumps to a new value at each threshold.

It rounds a number down to the nearest integer, like lowering it to the floor. This contrasts with the ceiling function ⌈x⌉, which rounds up to the nearest integer.

Step functions are typically introduced in Grade 11 Algebra 2 as part of the piecewise and special functions unit. They build on students' understanding of piecewise functions and domain restrictions.

Every step function is a piecewise function, but not every piecewise function is a step function. Step functions require that each piece be a constant (horizontal line). Other piecewise functions can have sloped or curved pieces.