Consider the following example:
This initial value problem as infinitely many solutions, because the Picard–Lindelöf theorem is not met. Indeed, $f'_x(t,x)$ is not defined for $x(0)=0$. What I have no idea is: Why $\phi_c(t)=0$ for $t \in (-\infty,c]$. Where $\phi_c(t)=(t-c)^3$ comes from?