I'm finding the notation really confusing so I am going to change it
x=Z+αwhere Z=⌊x⌋and0≤α<1
f(x)=f(Z+α)=Z−α
f(3.2)= 3-0.2=2.8
f(-3.2)=f(-4+0.8) =-4-0.8 = -4.8
So this function certainly isn't even of odd.
It is not continuous either.
The continuous segments are decreasing but I doubt that is valid, the graph as a whole is not decreasing.
anyway, that is something for you to think about.
Here is the graph
