Is the Greatest Integer Function Monotonically Increasing?

The greatest integer function is usually written $\lfloor x\rfloor$ and returns the largest integer less than or equal to $x$. Whether it is called “monotonically increasing” depends on the convention: some texts use that phrase to mean nondecreasing, while others reserve “increasing” for strictly increasing functions.

Under the nondecreasing definition, the answer is yes. Under the strict definition, the answer is no.

What the floor function does

For any real number $x$, $\lfloor x\rfloor$ is the integer $n$ satisfying

$$n\leq x<n+1.$$

Thus the function is constant on every interval $[n,n+1)$, where $n$ is an integer. For instance, every number from $2$ up to (but not including) $3$ has floor $2$:

$$\lfloor 2\rfloor=\lfloor 2.1\rfloor=\lfloor 2.9\rfloor=2.$$

At the next integer, the value jumps: $\lfloor 3\rfloor=3$.

Why it is nondecreasing

A function $f$ is nondecreasing if $x\leq y$ implies $f(x)\leq f(y)$. To check this for the floor function, suppose $x\leq y$, and let $m=\lfloor x\rfloor$. By the definition of floor,

$$m\leq x\leq y.$$

Since $m$ is an integer no greater than $y$, it cannot be larger than the greatest integer no greater than $y$. Therefore $m\leq\lfloor y\rfloor$, or

$$\lfloor x\rfloor\leq\lfloor y\rfloor.$$

This proves that the floor function is nondecreasing over all real numbers, including negative values.

A numerical example

Take $x=-1.4$ and $y=0.2$. Since $x<y$, compare their floors:

$$\lfloor-1.4\rfloor=-2, \qquad \lfloor0.2\rfloor=0.$$

The output has not gone down as the input increased: $-2\leq0$. The negative example also illustrates that floor means rounding down toward negative infinity, not rounding toward zero. In particular, $\lfloor-1.4\rfloor$ is $-2$, not $-1$.

Why it is not strictly increasing

A function is strictly increasing if $x<y$ always implies $f(x)<f(y)$. The floor function does not satisfy this condition. Choose two distinct numbers in the same unit interval, such as $2.1<2.9$. Then

$$\lfloor2.1\rfloor=2=\lfloor2.9\rfloor.$$

The inputs are strictly ordered, but the outputs are equal. In fact, the function is constant throughout each interval $[n,n+1)$, so there are many such counterexamples.

What happens at integers

The floor function never decreases as you move from left to right. It stays level between consecutive integers, then increases by one at each integer. For an integer $n$, values just below $n$ have floor $n-1$, while $\lfloor n\rfloor=n$. The function is right-continuous at integers, but it has a jump there.

This staircase shape is a useful way to distinguish monotonicity from continuity. A function can be monotone without being continuous, and it can be nondecreasing without being strictly increasing. The floor function has both of those features: it is nondecreasing and discontinuous at every integer.

Common mistakes

• Treating “increasing” as having only one meaning. Check whether a problem defines increasing as nondecreasing or strictly increasing. If it does not, state which interpretation you are using.
• Assuming monotone means strictly changing at every step. A nondecreasing function is allowed to stay constant over intervals.
• Rounding negative numbers toward zero. The greatest integer less than or equal to $-1.4$ is $-2$, so $\lfloor-1.4\rfloor=-2$.
• Confusing floor with ceiling. The ceiling function gives the least integer greater than or equal to its input; the floor gives the greatest integer less than or equal to it.

So, if “monotonically increasing” means nondecreasing, the greatest integer function is monotonically increasing. If the phrase means strictly increasing, it is not.