# Definition:Ceiling Function/Definition 2

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## Definition

Let $x \in \R$ be a real number.

The **ceiling function of $x$**, denoted $\ceiling x$, is defined as the smallest element of the set of integers:

- $\set {m \in \Z: x \le m}$

where $\le$ is the usual ordering on the real numbers.

## Also known as

The **ceiling function** is also known as the **least integer function**.

## Also see

- Set of Integers Bounded Below by Real Number has Smallest Element
- Smallest Element is Unique
- Equivalence of Definitions of Ceiling Function

## Technical Note

The $\LaTeX$ code for \(\ceiling {x}\) is `\ceiling {x}`

.

When the argument is a single character, it is usual to omit the braces:

`\ceiling x`

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Chapter $2$: Elementary Concepts of Graph Theory: $\S 2.2$: Isomorphic Graphs - 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