The Greatest integer function is defined as
In set notation we would write this as . It is also called the floor function and step function.
Domain of greatest integer function is
Graph of Greatest integer function
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
Here is given by
we can see that and
Hence is not one-one.
It is known that is always an integer. Thus there does not exists any element
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.
Geometrically, this means that there is no gap, split or missing point for
Greatest integer function is continous at all points apart from integral values of . It is also differentiable with
Since these limits cannot be equal to each other for any
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