Recently, I read a book ISSN 2195-9862 about the Filippov theory. There is a differential inclusion $$\dot{x}\in F(x)=\begin{cases} -1&x>0\\ [-1,1]&x=0\\ 1&x<0 \end{cases}\\ x(t_0)=x_0$$ For this differential inclusion, the authors write that "The Filippov solution is $$x(t)=\begin{cases} x(0)-sgn(x_0)t,\ t\in [0,|x(0)|]\\ 0, t\geq |x(0)| \end{cases}\\$$ where $sgn(x)$ is the single-valued mapping and $sgn(0)=0$."
My question is how to get this "Filippov solution"? I can follow the part $x(0)-sgn(x_0)t,\ t\in [0,|x(0)|]$. But why $x(t)=0$ for $t\geq |x(0)|$? I cannot figure out what happen at $x=0$.