AP Calculus AB and BC glossary

Greatest integer function

Also called: Floor function

The greatest integer function returns the largest integer less than or equal to x, so it is a staircase that steps up by one at every integer. It is the standard example of a function with a jump discontinuity at each integer, where the one-sided limits differ by exactly one.

⌊x⌋=max⁡{n∈Z:n≤x}\lfloor x \rfloor = \max\{n \in \mathbb{Z} : n \le x\}

At x=2x = 2 the left-hand limit is 11 and the right-hand limit is 22, so the limit does not exist. The function is right-continuous at each integer, because ⌊2⌋=2\lfloor 2 \rfloor = 2 matches the limit from the right.

The mistake

Treating it as rounding. It always steps down, so the value at negative one point five is negative two, not negative one.

Appears in: Unit 1: Limits and Continuity