can you please explain this z(x)=x-[x]

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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           = 0.6 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 - 7        = 0. 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.