The Greatest integer function is defined as , where denotes the greatest integer that is less than or equal to .
In set notation we would write this as . It is also called the floor function and step function.
Example. , and
Domain of greatest integer function is ( the set of all real numbers) and range is ( the set of all integers ) as it only attains integer value.
Graph of Greatest integer function
Let be the greatest integer function define as
Let us calculate some value of for different value of .
The graph of greatest integer function is given below.
Greatest integer function is neither one-one nor onto
A function from to is said to be one-one if whenever then . It means distinct elements have distinct image.
Here is given by ,
we can see that and .
Hence is not one-one.
A function from to is called onto if for all there is an such that
Now consider .
It is known that is always an integer. Thus there does not exists any element such that . Thus is not onto.
Hence , the Greatest integer function is neither one-one nor onto.
Continuity and differentiability of greatest integer function
Continuity and differentiability are properties of a function at a specific point rather than properties of a function as a whole . A function is said to be continuous at a point c if each of the following condition is satisfied.
(i) exists. (c is in the domain of )
(ii) exists and
Geometrically, this means that there is no gap, split or missing point for at and that a pencil could be moved along the graph of f(x) through without lifting it off the graph.
Greatest integer function is continous at all points apart from integral values of . It is also differentiable with , at all points apart from integral values of .
Let be an integer .Then we can find a sufficiently small real number such that whereas . Thus
Since these limits cannot be equal to each other for any , the function is discontinous at every integral points.
The greatest integer function is continuous at evety real no. other than integers. For example
Let’s take x=1.5 . Then
In general, if is any number which is not an integer and , is an integer , then .