In mathematics, the **floor function** is the function defined as follows: for a real number *x*, floor(*x*) is the largest integer less than or equal to *x*. For example, floor(2.3) = 2, floor(-2) = -2 and floor(-2.3) = -3. The floor function is also denoted by or .

We always have

*x*is an integer. For any integer

*k*and any real number

*x*, we have

*x*to the nearest integer can be expressed as floor(

*x*+ 0.5).

The floor function is not continuous, but it is upper semi-continuous.

A closely related mathematical function is the **ceiling function**,
which is defined as follows: for any given real number *x*, ceiling(*x*)
is the smallest integer no less than *x*. For example, ceiling(2.3) = 3,
ceiling(2) = 2 and ceiling(-2.3) = -2. The ceiling function is also denoted
by . It is easy to show the following:

*k*, we also have the following equality:

- .

*m*and

*n*are coprime positive integers, then