Page 129 - DMTH401_REAL ANALYSIS

Unit 9: Functions

The graph of the greatest integer function is shown in the Figure 9.17. Notes
Figure 9.17

Example: Prove that
[x + m] = [x] + m, " x R and m Z,
Solution: You know that for every x R, there exists an integer n such that
n < x < n + 1.

Therefore,
n + m < + m < n + 1 + m,
and hence
[x + m] = n + m = [x] + m,

which proves the result.

Task Test whether or not the function f : R  R defined by f(x) = x – [x] " x R, is periodic.
If it is so, find its period.

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