The function is $f(x) = \lfloor 1-x^2 \rfloor$.
$$f(x) = \left \{ \begin{array}{lr} -3 & : x \in [-2,-\sqrt{3})\\ -2 & : x \in [-\sqrt{3},-\sqrt{2})\\ -1 & : x \in [-\sqrt{2},-1)\\ 0 & : x\in [-1,1) \\ 1 & : x \in [1,\sqrt{2} )\\ 2 & : x \in [\sqrt{2}, \sqrt{3})\\ 3 & : x \in [\sqrt{3}, 2)\\ 4 & : x = 2 \end{array} \right.$$
But I don't know how to integrate that and I doesn't find examples.
Start with the piecewise definition of $F$ that you obtained for this problem. (Note, though, that you’ve calculated $f(x)$ incorrectly for $x\ge 1$: $f(1)=0$, and $f(x)$ is negative for all $x>1$.) As you can see, $F$ is piecewise linear, and adjacent pieces have different slopes. Thus, the derivative from the left at any of the ‘corners’ is different from the derivative from the right at that ‘corner’, and consequently $F$ is not differentiable there.