If `f(x) = [x] , 0<= {x} < 0.5 and f(x) = [x]+1 , 0.5<{x}<1 ` then prove that f (x) = -f(-x) (where[.] and{.} represent the greatest integer function and the fractional part function, respectively).

Explanation

`f(-x)={([-x]",",0 le {-x}lt0.5),([-x]+1",",0.5 lt {-x} lt1):}`
`={([-x]",",{-x}=0),([-x]",",0 lt {-x} lt 0.5),([-x]+1",",0.5 lt {-x} lt1):}`
`={(-[x]",",{x}=0),(-1-[x]",",0 lt 1-{x} lt 0.5),(-1-[x]+1",",0.5 lt 1-{x} lt1):}`
`={(-[x]",",{x}=0),(-1-[x]",",0.5 lt {x} lt 1),(-[x]",",0 lt {x} lt0.5):}`
`={(-[x]",",0 le{x}lt0.5),(-1-[x]",",0.5 lt {x} lt 1):}`
`= -f(x)`