Question

Prove that 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 1

f : R → R given by f(x) = [x].

(a) f(1.2) = 1, f(1.5) = 1

⇒ f is not one-one.

(b) All the images of x ∈ R belonging to its domain have integers as the images in codomain. But no fraction proper or improper belonging to codomain of f has any pre-image in its domain.

⇒ f is not onto.

Hence, f is neither one-one nor onto.