Greatest integer function
By: Ahmad Ghandehari
1-1.
means,
the greatest integer which is less than or equal to x.
For
each x belongs to the set of real numbers, and n belongs to the set of
integer numbers, we can write:
(1)
It means:
a):
and
b):
.
Examples:
a):
b):
c):
d):
e):
f ):
g):
Example 1:
If,
then
find the interval of x variation.
Solution:
1-2. For each n belongs to the set of integer numbers we can write.
(2)
Examples:
a)
b)
c)
Problem 1:
If,
,
then find the interval of x variation.
Solution:
.
Problem 2:
If ,
then
find the interval of x variation.
Solution:
Problem 3:
Supposing
and
,
then find the interval of y variation.
Solution:
.
Problem 4:
If ,
,
then find the interval of x variation.
Solution:
.
Problem 5:
How many
integer roots has the equation 
Solution:
We know that 0!=1!=1 , so
a)
b)
Problem 6:
Find the
amount of area which is restricts to the graph of
.
Solution:
We know that
and
so
a)
b)
and 
so
square
unit.
Problem 7:
Prove that
Solution:
Let
and
We want to
prove that
so
Now we have to prove that
a):let,
Therefore
b):let,
There fore
Problem 8:
If
,
then find the interval of x variation.
Solution:
We prove in problem 7;
,
so
.
Problem 9:
If,
,
then find the interval of x variation.
Solution:
,
therefore,
.
Problem 10:
If
,
then find all xs.
Solution:
Let
and
therefore 
.
Problem 11:
If
,
then find all xs.
Solution:
let
and
; 
; 
; 
.
Problem 12:
We know that
Now if
then
find the interval of x variation.
Solution:
a)
Let ,
therefore 
.
b)
Let 
therefore
so
x belongs to the set of
1-3. Greatest integer function properties
This is the graph of
when
1-
The domain of is
the set of real numbers.
2- The range of this function is the set of integer numbers.
3- In any integer numbers this function does not limit.
4-
If then
this function does not continuous and
does
not exists.
5-
If then
this function has limit and is continuous and differentiable and
6-
is
increasing function.