imc

There are four cases we need to consider here. Case 1: $x$ is a positive integer. Without loss of generality, assume $x=1$. Then $f(1)=1-1=0$. Case 2: $x$ is a positive fraction. Without loss of generality, assume $x=\frac{1}{2}$. Then $f\left(\frac{1}{2}\right)=0-0=0$. Case 3: $x$ is a negative integer. Without loss of generality, assume $x=-1$. Then $f(-1)=1-1=0$. Case 4: $x$ is a negative fraction. Without loss of generality, assume $x=-\frac{1}{2}$. Then $f\left(-\frac{1}{2}\right)=0-1=-1$. Thus the range of the function $f$ is $\boxed{\{-1,0\}}$.