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Question

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.

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Solution

f: RR is given by,

f(x) = [x]

It is seen 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, consider 0.7 ∈ R.

It is known that f(x) = [x] is always an integer. Thus, there does not exist any element xR such that f(x) = 0.7.

f is not onto.

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


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