Hello,

x=[x]+{x}, where 0<{x}<1 (the decimal part)

If {x}<1/2, then [x+1/2]=[x] and [2x]=[2[x]+2{x}]=[2[x]]=2[x].

So if {x}<1/2, we indeed have [x]+[x+1/2]=2[x]=[2x]

and if {x}>1/2, [x+1/2]=[x]+1 and [2x]=[2[x]+2{x}]. But 1<2{x}<2. So [2x]=[2[x]]+1=2[x]+1.

So if {x}>1/2, we indeed have [x]+[x+1/2]=2[x]+1=[2x]