Definition:Ceiling Function

Definition
Let $x \in \R$.

Then $\left \lceil {x} \right \rceil$ is defined as:
 * $\left \lceil {\cdot} \right \rceil: \R \to \Z: \left \lceil {x} \right \rceil = \inf \left({\left\{{m \in \Z: m \ge x}\right\}}\right)$

That is, $\left \lceil {x} \right \rceil$ is the smallest integer greater than or equal to $x$.

It immediately follows that:


 * $\left \lceil {x} \right \rceil$ is an integer
 * $\left \lceil {x} \right \rceil - 1 < x \le \left \lceil {x} \right \rceil < x + 1$
 * $\forall n \in \Z: \left \lceil {x + n} \right \rceil = \left \lceil {x} \right \rceil + n$

This is called the ceiling function.

Notation
The notation given here was introduced in the 1960s by and popularised by.

Compare the notation for the floor function.