jmc

The discriminant of the quadratic is $7^2-4(13)=-3<0$, so the quadratic has no real roots and is always positive for real inputs. The function is undefined if $0\le x^2-7x+13<1$, since $\lfloor x^2-7x+13\rfloor =0$ in that case. Since the quadratic is always positive, we consider the inequality $x^2-7x+13<1$.

To find when $x^2-7x+13=1$, subtract 1 from both sides to obtain $x^2-7x+12=0$ and factor as $(x-3)(x-4)=0$, so $x=3$ or $x=4$. The parabola $x^2-7x+12$ is negative between these points, so we must exclude the interval $(3,4)$ from the domain. So the domain of $f$ is $\boxed{(-\infty ,3]\cup [4,\infty )}$.