# Characteristics of Floor and Ceiling Function

## Theorem

Let $f: \R \to \Z$ be an integer-valued function which satisfies both of the following:

- $(1): \quad \map f {x + 1} = \map f x + 1$
- $(2): \quad \forall n \in \Z_{> 0}: \map f x = \map f {\dfrac {\map f {n x} } n}$

Then either:

- $\forall x \in \Q: \map f x = \floor x$

or:

- $\forall x \in \Q: \map f x = \ceiling x$

### Real Domain

Let $f: \R \to \Z$ be an integer-valued function which satisfies both of the following:

- $(1): \quad \map f {x + 1} = \map f x + 1$
- $(2): \quad \forall n \in \Z_{> 0}: \map f x = \map f {\dfrac {\map f {n x} } n}$

Then it is not necessarily the case that either:

- $\forall x \in \R: \map f x = \floor x$

or:

- $\forall x \in \R: \map f x = \ceiling x$

## Proof

From $(1)$, by induction we have:

- $\forall n \in \N: \map f {x + n} = \map f x + n$

and

- $\forall n \in\N: \map f {x - n} = \map f x - n$

and therefore, in particular:

- $(3): \quad \forall n \in \Z: \map f n = \map f 0 + n$

From $(2)$, we get

\(\displaystyle \map f 0\) | \(=\) | \(\displaystyle \map f {\map f 0}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f 0 + \map f 0\) | using $(3)$ and $f$ being integer valued |

Hence

- $\map f 0 = 0$

Thus from $(3)$ it follows that:

- $\forall n \in \Z: \map f n = n$

Suppose that $\map f {\dfrac 1 2} = k \le 0$.

Then:

\(\displaystyle k\) | \(=\) | \(\displaystyle \map f {\dfrac 1 {1 - 2 k} \map f {\dfrac 1 2 - k} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f {\dfrac 1 {1 - 2 k} \paren {\map f {\dfrac 1 2} - k} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f 0\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) |

We show that in this case, by induction:

- $\map f {\dfrac 1 n} = 0$ for all $n \in \N$

Induction hypothesis:

- $\map f {\dfrac 1 {n - 1} } = 0$

Then from $(1)$:

- $\map f {\dfrac n {n - 1} } = \map f {\dfrac 1 {n - 1} + 1} = 0 + 1 = 1$

so:

\(\displaystyle \map f {\dfrac 1 n}\) | \(=\) | \(\displaystyle \map f {\dfrac 1 n \map f {\dfrac n {n - 1} } }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f {\dfrac 1 {n - 1} }\) | using $(2)$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) |

Finally, we show by induction on $m$ that even:

- $\map f {\dfrac m n} = 0$

for $m \in \set {1, \ldots, n - 1}$.

Above we have shown this for $m = 1$.

Let $1 \le m < n$.

If $m \divides n$, then

- $\map f {\dfrac m n} = \map f {\dfrac 1 {n / m} } = 0$

Otherwise, write:

- $n = \paren {k - 1} m + r$

Then:

- $k > 1$

and:

- $1 \le r \le m - 1$

We therefore have:

- $k m = n + m - r$

so by the induction hypothesis:

- $\map f {\dfrac {k m} n} = \map f {1 + \dfrac {m - r} n} = 1 + \map f {\dfrac {m - r} n} = 1$

Then:

\(\displaystyle \map f {\dfrac m n}\) | \(=\) | \(\displaystyle \map f {\dfrac 1 m \map f {\dfrac {m k} n} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f {\dfrac 1 m}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) |

By $(1)$, this shows that

- $\map f {\dfrac 1 2} \le 0 \implies \map f x = \floor x$

for all rational $x$.

Suppose that $\map f {\dfrac 1 2} = k > 0$.

Then the integer-valued function $g: \R \to \Z$ satisfies:

- $\map g x = -\map f {-x}$

satisfies $(1)$ and $(2)$, and also:

\(\displaystyle \map g {\dfrac 1 2}\) | \(=\) | \(\displaystyle 1 - \map f {\dfrac 1 2}\) | |||||||||||

\(\displaystyle \) | \(\ge\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map f x\) | \(=\) | \(\displaystyle -\map g {-x}\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\floor {-x}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \ceiling x\) | Floor of Negative equals Negative of Ceiling |

Thus:

- $\map f {\dfrac 1 2} > 0 \implies \map f x = \ceiling x$

for all rational $x$.

$\blacksquare$

## Historical Note

The result *Characteristics of Floor and Ceiling Function* was presented by Peter Eisele and Karl Peter Hadeler in $1990$.

The previous work done in $1905$ by Georg Hamel to establish that the result does not necessarily hold in the real domain is to be noted.

## Sources

- 1990: P. Eisele and K.P. Hadeler:
*Games of cards, dynamical systems, and a characterization of the floor and ceiling functions*(*Amer. Math. Monthly***Vol. 97**: 466 – 477) www.jstor.org/stable/2323829

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $49$