Discussion
The ceiling function of a real number \(x\), denoted by \(ce(x)\), is defined as the smallest integer that is greater than or equal to \(x\). Similarly, the floor function, denoted by \(fl(x)\), is defined as the largest integer that is smaller than or equal to \(x\). Which one of the following statements is NOT correct for all possible values of \(x\)?
Answer: D
The statement that fails for some \(x\) is \(fl(x) \lt ce(x)\).
When \(x\) is an integer, both the ceiling and the floor equal \(x\) itself, so \(fl(x) = ce(x)\) and the strict inequality is false. For example, \(fl(3) = ce(3) = 3\).
The other three hold for every real \(x\): by definition the ceiling is at least \(x\) and the floor is at most \(x\), and so \(ce(x) \ge x \ge fl(x)\).
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