The ceiling function (also known as the least integer function) of a real number denoted is defined as the smallest integer that is not smaller than
In general, is the unique integer satisfying
The ceiling is related to the floor function by the formula
What is the range of that satisfies
Let , where is an integer by the definition of the ceiling function. Then
(1) for any integer
These can all be proved from the analogous properties for the floor function.
As with the floor function, it is often easiest to write where is an integer and
Find all solutions to
Write as above. Then or depending on In the first case, where the equation becomes which has no solution. In the second case, where the equation becomes so or The only integer solution is .
So the range of solutions is the interval
As with floor functions, the best strategy with integrals or sums involving the ceiling function is to break up the interval of integration (or summation) into pieces on which the ceiling function is constant.
This is clearly Now break the interval of integration up into pieces on which This becomes