The critical points of the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$ are:

Critical points include points where $f^{\prime }(x)=0$ and points where $f(x)$ or $f^{\prime }(x)$ does not exist.$f(x)=\frac{x^2}{x^2-1}$ and $f^{\prime }(x)=\frac{-2x}{(x^2-1{)}^2}$.$f^{\prime }(x)=0$ at $x=0$.$f(x)$ and $f^{\prime }(x)$ are not defined at $x=\pm 1$.Therefore, the critical points are $x=-1,0,1$.