This is a variation of the greatest integer function. This particular problem says: the function of x is equal to x minus the greatest integer less than or equal to x.
This is easier to see with an example. For instance, if x = 5.6, then...
z(5.6) = 5.6 - [[5.6]]
= 5.6 - 5
It's only shows differing results when dealing with non-integers. If you plug an integer in, you'll just get 0.
For instance, if x = 7, then...
z(7) = 7 - []
= 7 - 7
The differentiation only comes when you have to "change" the number inside the brackets to the greatest integer less than the value of x. If an integer is used, 0 is the outcome.