Q. 33.8( 61 Votes )

Prove that the Greatest Integer Function f : R R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Answer :

It is given f : R R, given by f (x) = [x]

We can see that f(1.2) = [1.2] = 1


f(1.9) = [1.9] = 1


f(1.2) = f(1.9), but 1.2 1.9.


f is not one- one.


Now, let us consider 0.6 ϵ R.


We know that f(x) = [x] is always an integer.


there does not exist any element x ϵ R such that f(x) = 0.6


f is not onto.


Therefore, the greatest integer function is neither one-one nor onto.


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