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.

Question

Philoid · Accepted 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.