Graph of the Floor Function $y = \lfloor x \rfloor$

$y = \lfloor x \rfloor$ returns the largest integer not exceeding $x$¹. It is called the floor function, and it is easiest to picture as the first integer you meet looking to the left from $x$ along the number line.

Domain and range

The domain is all real numbers. The value returned is always an integer, so the range is the set of all integers, a discrete collection rather than a continuous line. Only integer heights are ever taken.

Shape of the graph

The graph is a staircase. Each step is a horizontal segment of length $1$, and the value jumps up by $1$ at every integer.

Each step includes its left end and excludes its right end. On $1 \leq x < 2$ the value is always $1$, and the moment $x$ reaches $2$ it jumps to $2$.

Points of discontinuity

The function is discontinuous at every integer. Approaching an integer $n$ from the right the limit is $n$, while from the left it is $n - 1$, and the two disagree. The size of the jump is always $1$.

At non-integer points the value is constant nearby, so the function is differentiable there with derivative $0$. It is made of nothing but flat steps of slope $0$ and sudden jumps.

Monotonicity

The value never decreases as $x$ increases, so the function is non-decreasing. It is not strictly increasing, however, since the value stays put along each step.

A caution with negative numbers

Note that $\lfloor -1.2 \rfloor = -2$, not $-1$. The floor function does not discard the fractional part; it always rounds toward $-\infty$. That differs from the integer division of many programming languages, which rounds toward $0$, so negative numbers deserve care.

Relations to other functions

• The ceiling function $\lceil x \rceil$, returning the smallest integer at least $x$, is its counterpart
• The difference $x - \lfloor x \rfloor$ is the fractional part, a sawtooth wave of period $1$
• Rounding to the nearest integer can be written $\lfloor x + 0.5 \rfloor$

Applications

The number of digits of a positive integer $N$ is found with the floor function.

The quotient of an integer division $\left\lfloor \dfrac{a}{b} \right\rfloor$, the computation of array indices and the paging of a list all use it: wherever a continuous quantity has to be cut into steps.

1. Floor and ceiling functions, Wikipedia