Is the Greatest Integer Function f: R → R Defined by f(x) = [x] One-One and Onto?

No, the greatest integer function is neither one-one (injective) nor onto (surjective).

Why Not One-One: The greatest integer function, also called the floor function, maps multiple real numbers to the same integer.

Example:

• f(2.1) = 2
• f(2.5) = 2
• f(2.9) = 2

All values in [2, 3) map to 2, violating the one-one condition where different inputs must produce different outputs.

Why Not Onto: The codomain is R (all real numbers), but the range contains only integers {..., -2, -1, 0, 1, 2, ...}.

Non-integer real numbers like 2.5, π, or √2 are never outputs, so the function doesn't cover the entire codomain.

Key Takeaways:

• Many-to-one function (fails injectivity test)
• Range ≠ Codomain (fails surjectivity test)
• Neither injective nor surjective = not bijective