Definition:Ceiling Function/Definition 1

Definition

Let $x$ be a real number.

The ceiling function of $x$ is defined as the infimum of the set of integers no smaller than $x$:

$\ceiling x := \inf \set {m \in \Z: x \le m}$

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

Also known as

The ceiling function is also known as the least integer function or lowest integer function.

Also see

Theorems used in this definition:

• Continuum Property
• Infimum 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