# Characteristics of Floor and Ceiling Function/Real Domain

## Contents

## 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 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

Let $h: \R \to \R$ be a real function such that for all $x, y \in \R$:

\((3):\quad\) | \(\displaystyle \map h 1\) | \(=\) | \(\displaystyle 1\) | ||||||||||

\((4):\quad\) | \(\displaystyle \map h x + \map h y\) | \(=\) | \(\displaystyle \map h {x + y}\) |

Consider the integer-valued function $f: \R \to \Z$ defined as:

- $\map f x = \floor {\map h x}$

We claim that $f$ satisfies $(1)$ and $(2)$.

Proof for $(1)$:

We have that:

- $\map h {x + 1} = \map h x + \map h 1 = \map h x + 1$

by $(4)$ and $(3)$.

It follows that:

\(\displaystyle \map f {x + 1}\) | \(=\) | \(\displaystyle \floor {\map h {x + 1} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \floor {\map h x + 1}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \floor {\map h x} + 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f x + 1\) |

Thus $h$ satisfies $(1)$.

$\Box$

Proof for $(2)$:

Since $h$ satisfies $(4)$, it is an additive function.

By $(3)$ and since Additive Function is Linear for Rational Factors, this implies

- $(5): \quad \map h x = x$

for all $x \in \Q$.

Let $x \in \R$.

Define $\alpha$ and $\beta$ by

- $(6): \quad \alpha := \floor {\map h x} = \map f x$
- $(7): \quad \beta := \map h x - \alpha$

Then:

\((8):\quad\) | \(\displaystyle 0\) | \(\le\) | \(\displaystyle \beta < 1\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 0\) | \(\le\) | \(\displaystyle n \beta < n\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 0\) | \(\le\) | \(\displaystyle \floor {n \beta} \le n - 1\) |

and:

\(\displaystyle \map f {\dfrac {\map f n x} n}\) | \(=\) | \(\displaystyle \floor {\map h {\dfrac {\floor {\map h {n x} } } n} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \floor {\dfrac {\floor {n \map h x} } n}\) | $(5)$, Additive Function is Linear for Rational Factors | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \floor {\dfrac {\floor {n \paren {\alpha + \beta} } } n}\) | $(6)$, $(7)$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \floor {\dfrac {n \alpha + \floor {n \beta} } n}\) | as $\alpha \in \Z$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \alpha + \floor {\dfrac 1 n \floor {n \beta} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \floor {\map h x}\) | $(8)$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f x\) | Definition of $\map f x$ |

Thus $h$ satisfies $(2)$.

$\Box$

We have that:

Thus we can consider $\R$ as a vector space over $\Q$.

We also have that Square Root of 2 is Irrational

Hence the set $\set {1, \sqrt 2}$ is a linearly independent set in the vector space $\R$.

From Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set:

- there exists a basis $B$ of $\R$ which includes $1$ and $\sqrt 2$.

Then each $x \in \R$ can be written as a finite sum:

- $x := \displaystyle \sum_{i \mathop = 1}^n b_i x_i$

where $b_i \in B$, $x_i \in \Q$ and $n$ depends on $x$.

Let $f$ be defined as:

- $\map f x = \displaystyle \sum_{i \mathop = 1}^n \map f {b_i} x_i$

From Expression of Vector as Linear Combination from Basis is Unique, we have:

- $\map f x + \map f y = \map f {x + y}$

no matter how $\map f b$ is defined for $b \in B$.

Let $f$ be further defined as:

- $\map f 1 = 1$

and, for example:

- $\map f {\sqrt 2} = 4$

Then $f$ satisfies $(1)$ and $(2)$.

But:

- $\map f {\sqrt 2} \notin \set {1, 2}$

$\blacksquare$

## Historical Note

The limitation of Characteristics of Floor and Ceiling Function to rational numbers was demonstrated by Georg Karl Wilhelm Hamel in $1905$.

## Sources

- 1905: Georg Hamel:
*Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x+y)=f(x)+f(y)$*(*Math. Ann.***Vol. 60**: 459 – 462) - 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 $49$