# Greatest integer function (floor value function)

*y*=

*f*(

*x*) = [

*x*] (Greatest integer â‰¤

*x*)

Domain â†’

*R*; Range â†’*I*;# Properties of greatest integer function

- [
*x*] =*n*(*n*âˆˆ*I*) â‡’*x*âˆˆ [*n*,*n*+1) -
*x*â€“ 1 < [*x*] â‰¤*x* - [â€“
*x*] + [*x*] = - [
*x*] â‰¥*n*â‡’*x**â‰¥**n*,*n*âˆˆ*I* - [
*x*] â‰¤*n*â‡’*x*<*n*+ 1,*n*âˆˆ*I* - [
*x*] >*n*â‡’*x*â‰¥*n*+ 1,*n*âˆˆ*I**x*] â‰¥ 2 â‡’*x*âˆˆ [2, âˆž)*x*] > 3 â‡’ [*x*] â‰¥ 4 â‡’*x*âˆˆ [4, âˆž)*x*] â‰¤ 3 â‡’*x*âˆˆ (â€“âˆž, 4)

**Fractional part function**

*y*=

*f*(

*x*) = {

*x*} =

*x*â€“ [

*x*]

Domain â†’
[
[
{
{

*R*; Range â†’ [0, 1); Period â†’ 1;*x*+*y*] = [*x*] + [*y*], if 0 â‰¤ {*x*} + {*y*} < 1*x*+*y*] = [*x*] + [*y*] + 1 , 1 â‰¤ {*x*} + {*y*} < 2*x*} + {â€“*x*} = 0 if*x*âˆˆ*I**x*} + {â€“*x*} = 1 if*x*âˆ‰ I