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?

The correct option is fl(x) < ce(x).

To understand why this is the case, let us analyze the definitions of the floor and ceiling functions and evaluate each statement.

1. Definition of Ceiling Function:
The ceiling function, denoted by ce(x), is the smallest integer greater than or equal to x. By definition, this means:

$\mathrm{ce}\left(x\right)\ge x$

Since this inequality holds true for all real numbers x, the statement ce(x) ≥ x is correct.

2. Definition of Floor Function:
The floor function, denoted by fl(x), is the largest integer less than or equal to x. By definition, this means:

$\mathrm{fl}\left(x\right)\le x$

Since this inequality holds true for all real numbers x, the statement fl(x) ≤ x is correct.

3. Comparing ceiling and floor:
Combining the two inequalities above, we have:

$\mathrm{fl}\left(x\right)\le x\le \mathrm{ce}\left(x\right)$

This directly implies:

$\mathrm{fl}\left(x\right)\le \mathrm{ce}\left(x\right)$

Which can also be written as:

$\mathrm{ce}\left(x\right)\ge \mathrm{fl}\left(x\right)$

Therefore, the statement ce(x) ≥ fl(x) is correct for all real numbers x.

4. Analyzing the incorrect statement:
The statement fl(x) < ce(x) asserts that the floor of x is strictly less than the ceiling of x for all real numbers x. Let us test this when x is an integer, for example, x = 3.

The floor of an integer is the integer itself:

$\mathrm{fl}\left(3\right)=3$

The ceiling of an integer is also the integer itself:

$\mathrm{ce}\left(3\right)=3$

Comparing them, we get:

$\mathrm{fl}\left(3\right)=\mathrm{ce}\left(3\right)=3$

Here, the strict inequality does not hold because 3 is equal to 3, not strictly less than 3. Therefore, the statement fl(x) < ce(x) is false whenever x is an integer. This makes it NOT correct for all possible values of x.