Range of Values of Ceiling Function

Theorem

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

Let $\left \lceil{x}\right \rceil$ be the ceiling of $x$.

Let $n \in \Z$ be an integer.

Then the following results apply:

Number greater than Integer iff Ceiling greater than Integer

$\ceiling x > n \iff x > n$

Number not greater than Integer iff Ceiling not greater than Integer

$\ceiling x \le n \iff x \le n$

Integer equals Ceiling iff between Number and One More

$\ceiling x = n \iff x \le n < x + 1$

Integer equals Ceiling iff Number between Integer and One Less

$\ceiling x = n \iff n - 1 < x \le n$

Also see

• Range of Values of Floor Function