Function *f* from \(\mathbb{R}\) to \(\mathbb{Z}\) such that for every real number *x*, *f*(*x*) is equal to the greatest integer less than or equal to *x*.

Synonym of greatest integer function.

### Notation

- The
*integer part*of*x*is denoted by [*x*]. - The relationship defined by
*f*(*x*) = [*x*] defines the**basic model**of the floor function.

### Properties

- The general model, which is translated by a step function is defined by the relation
*f(x)*= a[b*x*] , in which parameter**a**characterizes a “riser” (or vertical jump) on the graph, and parameter**b**characterizes the length of a horizontal step or each horizontal segment of the graph. - Parameter
**a**causes a vertical scale change of value |a| and affects the orientation—or steepness—of the graph depending on whether**a**is positive or negative. - Parameter
**b**causes a horizontal scale change of value \(\left| \dfrac {1}{b} \right|\) and affects the orientation – or steepness – of the graph, depending on whether**b**is positive or negative. - The standard form of the general model is\(f(x) = \textrm{a}\left[\textrm{b}\left(x – \textrm{h}\right)\right] + \textrm{k}\) where parameters
*h*and*k*characterize a horizontal translation and a vertical translation, respectively, of the graph of the basic relation.

### Example

The graph of the function defined by the rule *f*[*x*] is that of a step function.

The following graph represents the function defined by the relation *f*[*x*] = −3[0,5*x*].

In this case, the direction of the graph was reversed, as parameter **a** is negative. The graph underwent a scale change of 3 units along the vertical axis and a scale change of |2| units along the horizontal axis.