I'm studying the book Lições de Equações Diferenciais Ordinárias by Jorge Sotomayor. There is a proof of a theorem about maximum solutions of first order ODE in page 17 where I can't understand a detail. If I assume that the function is locally Lipschitz the problem goes away but the book treats a slightly more general case. So I was wondering if the following theorem is true.
Let $\Omega\subseteq\mathbb{R}\times\mathbb{R}^n$ be an open set, $(t_0,x_0)$ an element of $\Omega$ and $f:\Omega\to \mathbb{R}^n$ a continuous function. Suppose there exists an open interval $I$ such that $x'=f(t,x), x(t_0)=x_0$ has a unique solution in $I$. Suppose $J$ is an open subinterval of $I$ containing $t_0$. Does the equation $x'=f(t,x), x(t_0)=x_0$ have a unique solution in $J$?
If the previous theorem is not true, assume that the hypothesis holds for every $(t_0,x_0)\in \Omega$, i.e. that for each $(t_0',x_0')\in \Omega$ there exists an open interval $I(t_0',x_0')$ such that the $x'=f(t,x), x(t_0')=x_0'$ has a unique solution in $I(t_0,x_0)$, does the conclusion hold now?
This is the actual theorem in Sotomayor's book
Let $f$ be continuous in an open set $\Omega\subseteq \mathbb{R}\times\mathbb{R}^n$. Suppose that for each $(t_0,x_0)\in \Omega$ there exists a unique solution of $x'=f(t,x),x(t_0)=x_0$ defined in an open interval $I=I(t_0,x_0)$ (for example, if $f$ is locally Lipschitz this condition is satisfied). Then, for each $(t_0,x_0)\in \Omega$ there exists a unique solution $\varphi=\varphi(t,t_0,x_0)$ of $x'=f(t,x),x(t_0)=x_0$, defined in an open interval $M(t_0,x_0)=(w_-(t_0,x_0),w_+(t_0,x_0))$ with the property that every solution $\psi$ of $x'=f(t,x),x(t_0)=x_0$ in an interval $I$ satisfies $I\subseteq M(t_0,x_0)$ and $\psi=\varphi|I$