Consider the following scalar ODE: for any initial data $x(0)=:x_0\in(a,b)\subset [0,1]$, \begin{equation} \dot{x}(t)= \begin{cases} 1-x, & \text{if } x<c\\ 0, & \text{if } x=c \\ -x, & \text{if } x>c \end{cases}\\ \end{equation} where $c\in (a,b)$. We seek for $\textbf{Caratheodory's solution}$ for all $t\geq 0$.
For existence:
i): If $x(0)\in (a,c)$: I found a solution being $x(t)=1-(1-x_0)e^{-t}$ for $t<\tilde{t}$ and $x(t)=c$ for $t\geq \tilde{t}$ where $\tilde{t}$ is the time $x(t)$ reaches $c$ and note that $x(t)$ is strictly increasing in $(a,c)$;
ii): If $x(0)=0$, a solution is $X(t)=0$ for all $t\geq 0$;
iii): If $x(0)\in (c,b)$, a solution is $x(t)=x_0e^{-t}$ for $t\leq \hat{t}$ and $x(t)=c$ for all $t\geq \hat{t}$, where $\hat{t}$ is the time $x(t)$ reachese $c$. Note that $x(t)$ is strictly decreasing while it's in $(c,b)$.
However, my main concern is the $\textbf{uniqueness}$ of the Caratheodory solution to this ODE. First of all, are these solutions unique? If not, any mild conditions we can impose to restore uniqueness?