Why are the maximal interval's endpoints semicontinuous functions of the initial condition?

Question

Consider the solution of the autonomous initial value problem

$$x' = f(x), \quad x(0) = x_0, \quad f \in C^1$$

as a function of $x_0$. Let $L(x_0)$ and $U(x_0)$ be the lower and upper endpoints of the maximal interval where the solution is defined. How do I show that $L$ and $U$ are upper and lower semicontinuous functions of $x_0$, respectively?