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.

F : R → R given by f(x) = [x] 

Injectivity : Let  x₁ = 2.5 and x₂ = 2 be two elements of R

f(x₁) = f(2.5) = [2.5] = 2     

f(x₂) = f(2) = [2] = 2    

∴ f(x₁) = f(x₂) for x₁ ≠ x₂ 

⇒ f(x) = [x]  

Subjectivity : Let y = 2.5 ∈ R be any element 

∴ f(x) = 2.5   

⇒ [x] = 2.5

which is not possible as [x] is always an integer. 

∴ f(x) = [x] is not onto.