The Function [x]

For example,

[7.362] = 7, [π] = 3, [-2.43] = -3.

For each integer n, [n] is equal to n. The function [x] is continuous when x is not an integer but is discontinuous when x is an integer n. At an integer n, both one-sided limits exist but are different,

lim_x→n- f(x) = n - 1, lim_x→n+ f(x) = n.

The graph of [x] looks like a staircase. It has a step, or jump discontinuity, at each integer n. The function [x] will be useful in the last section of this chapter. Some hand calculators have a key for either the greatest integer function or for the similar function that gives [x] for positive x and [x] + 1 for negative x.