Y= floor(x)-x+2

from the graph, it is periodic. but how to prove it?

It is sufficient to prove that is periodic. We claim that the period is 1, and here's the proof:

Let

by definition of periodic function, we are claiming that, for every value of x,

The floor is additive with integers: if a is a real number and b is an integer, we have

So, we have

Which proves that the function is periodic with period 1. The idea is that is the decimal part of x. For example, if you start at x=2.5 you have

If you go one period ahead, i.e. if you add 1, you have

And as you can see, the integer part doesn't matter, because you'll always have a difference in the form

xx - xx.5 = -0.5