# Properties of Ceiling Function

## Theorem

### Number is between Ceiling and One Less

$\ceiling x - 1 < x \le \ceiling x$

### Ceiling is between Number and One More

$x \le \ceiling x < x + 1$

where $\ceiling x$ is the ceiling of $x$.

### Ceiling of Number plus Integer

$\forall n \in \Z: \left \lceil {x} \right \rceil + n = \left \lceil {x + n} \right \rceil$

### Ceiling minus Real Number

$\forall x \in \R: \left \lceil {x} \right \rceil - x \in \left[{0 \,.\,.\, 1}\right)$

### Real Number is between Ceiling Functions

$\forall x \in \R: \left \lceil {x - 1} \right \rceil \le x < \left \lceil {x} \right \rceil$

### Real Number is Ceiling minus Difference

Let $n$ be a integer.

The following are equivalent:

$(1): \quad$ There exists $t \in \hointr 0 1$ such that $x = n - t$
$(2): \quad n = \ceiling x$

### Ceiling Function is Idempotent

$\ceiling {\ceiling x} = \ceiling x$

## Range of Values of Ceiling Function

### Number greater than Integer iff Ceiling greater than Integer

$\left \lceil{x}\right \rceil > n \iff x > n$

### Number not greater than Integer iff Ceiling not greater than Integer

$\left \lceil{x}\right \rceil \le n \iff x \le n$

### Integer equals Ceiling iff between Number and One More

$\left \lceil{x}\right \rceil = n \iff x \le n < x + 1$

### Integer equals Ceiling iff Number between Integer and One Less

$\left \lceil{x}\right \rceil = n \iff n - 1 < x \le n$